UNITED STATES
                       SECURITIES AND EXCHANGE COMMISSION
                             Washington, D.C. 20549

                                   FORM N-CSR

        CERTIFIED SHAREHOLDER REPORT OF REGISTERED MANAGEMENT INVESTMENT
                                   COMPANIES

                  Investment Company Act file number 811-22528
                                                    -----------

                     First Trust Energy Infrastructure Fund
               (Exact name of registrant as specified in charter)

                       120 East Liberty Drive, Suite 400
                               Wheaton, IL 60187
         --------------------------------------------------------------
              (Address of principal executive offices) (Zip code)

                             W. Scott Jardine, Esq.

                          First Trust Portfolios L.P.
                       120 East Liberty Drive, Suite 400
                               Wheaton, IL 60187
         --------------------------------------------------------------
                    (Name and address of agent for service)

        registrant's telephone number, including area code: 630-765-8000
                                                           --------------

                      Date of fiscal year end: November 30
                                              -------------

                     Date of reporting period:   May 31, 2016
                                              -------------------


Form N-CSR is to be used by management investment companies to file reports with
the Commission not later than 10 days after the transmission to stockholders of
any report that is required to be transmitted to stockholders under Rule 30e-1
under the Investment Company Act of 1940 (17 CFR 270.30e-1). The Commission may
use the information provided on Form N-CSR in its regulatory, disclosure review,
inspection, and policymaking roles.

A registrant is required to disclose the information specified by Form N-CSR,
and the Commission will make this information public. A registrant is not
required to respond to the collection of information contained in Form N-CSR
unless the Form displays a currently valid Office of Management and Budget
("OMB") control number. Please direct comments concerning the accuracy of the
information collection burden estimate and any suggestions for reducing the
burden to Secretary, Securities and Exchange Commission, 100 F Street, NE,
Washington, DC 20549. The OMB has reviewed this collection of information under
the clearance requirements of 44 U.S.C. ss. 3507.





ITEM 1. REPORTS TO STOCKHOLDERS.

The Report to Shareholders is attached herewith.


                                  FIRST TRUST
                                 --------------
                                  E N E R G Y
                                 INFRASTRUCTURE
                                 --------------
                                      FUND
                                     (FIF)

                                                              SEMI-ANNUAL REPORT
                                                        FOR THE SIX MONTHS ENDED
                                                                    MAY 31, 2016

                                                                     FIRST TRUST

                                                     ENERGY INCOME PARTNERS, LLC





--------------------------------------------------------------------------------
TABLE OF CONTENTS
--------------------------------------------------------------------------------

                  FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
                               SEMI-ANNUAL REPORT
                                  MAY 31, 2016

Shareholder Letter..........................................................   1
At a Glance.................................................................   2
Portfolio Commentary........................................................   3
Portfolio of Investments....................................................   5
Statement of Assets and Liabilities.........................................  10
Statement of Operations.....................................................  11
Statements of Changes in Net Assets.........................................  12
Statement of Cash Flows.....................................................  13
Financial Highlights........................................................  14
Notes to Financial Statements...............................................  15
Additional Information......................................................  23

                  CAUTION REGARDING FORWARD-LOOKING STATEMENTS

This report contains certain forward-looking statements within the meaning of
the Securities Act of 1933, as amended, and the Securities Exchange Act of 1934,
as amended. Forward-looking statements include statements regarding the goals,
beliefs, plans or current expectations of First Trust Advisors L.P. ("First
Trust" or the "Advisor") and/or Energy Income Partners, LLC ("EIP" or the
"Sub-Advisor") and their respective representatives, taking into account the
information currently available to them. Forward-looking statements include all
statements that do not relate solely to current or historical fact. For example,
forward-looking statements include the use of words such as "anticipate,"
"estimate," "intend," "expect," "believe," "plan," "may," "should," "would" or
other words that convey uncertainty of future events or outcomes.

Forward-looking statements involve known and unknown risks, uncertainties and
other factors that may cause the actual results, performance or achievements of
First Trust Energy Infrastructure Fund (the "Fund") to be materially different
from any future results, performance or achievements expressed or implied by the
forward-looking statements. When evaluating the information included in this
report, you are cautioned not to place undue reliance on these forward-looking
statements, which reflect the judgment of the Advisor and/or Sub-Advisor and
their respective representatives only as of the date hereof. We undertake no
obligation to publicly revise or update these forward-looking statements to
reflect events and circumstances that arise after the date hereof.

                          MANAGED DISTRIBUTION POLICY

The Board of Trustees of the Fund has approved a managed distribution policy for
the Fund (the "Plan") in reliance on exemptive relief received from the
Securities and Exchange Commission that permits the Fund to make periodic
distributions of long-term capital gains as frequently as monthly each tax year.
Under the Plan, the Fund currently intends to continue to pay a recurring
monthly distribution in the amount of $0.11 per Common Share that reflects the
distributable cash flow of the Fund. A portion of this monthly distribution may
include realized capital gains. This may result in a reduction of the long-term
capital gain distribution necessary at year end by distributing realized capital
gains throughout the year. The annual distribution rate is independent of the
Fund's performance during any particular period. Accordingly, you should not
draw any conclusions about the Fund's investment performance from the amount of
any distribution or from the terms of the Plan. The Board of Trustees may amend
or terminate the Plan at any time without prior notice to shareholders.

                        PERFORMANCE AND RISK DISCLOSURE

There is no assurance that the Fund will achieve its investment objective. The
Fund is subject to market risk, which is the possibility that the market values
of securities owned by the Fund will decline and that the value of the Fund
shares may therefore be less than what you paid for them. Accordingly, you can
lose money by investing in the Fund. See "Risk Considerations" in the Additional
Information section of this report for a discussion of certain other risks of
investing in the Fund.

Performance data quoted represents past performance, which is no guarantee of
future results, and current performance may be lower or higher than the figures
shown. For the most recent month-end performance figures, please visit
http://www.ftportfolios.com or speak with your financial advisor. Investment
returns, net asset value and common share price will fluctuate and Fund shares,
when sold, may be worth more or less than their original cost.

The Advisor may also periodically provide additional information on Fund
performance on the Fund's webpage at http://www.ftportfolios.com.

                            HOW TO READ THIS REPORT

This report contains information that may help you evaluate your investment in
the Fund. It includes details about the Fund and presents data and analysis that
provide insight into the Fund's performance and investment approach.

By reading the portfolio commentary by the portfolio management team of the
Fund, you may obtain an understanding of how the market environment affected the
Fund's performance. The statistical information that follows may help you
understand the Fund's performance compared to that of relevant market
benchmarks.

It is important to keep in mind that the opinions expressed by personnel of EIP
are just that: informed opinions. They should not be considered to be promises
or advice. The opinions, like the statistics, cover the period through the date
on the cover of this report. The material risks of investing in the Fund are
spelled out in the prospectus, the statement of additional information, this
report and other Fund regulatory filings.





--------------------------------------------------------------------------------
SHAREHOLDER LETTER
--------------------------------------------------------------------------------

                  FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
                  SEMI-ANNUAL LETTER FROM THE CHAIRMAN AND CEO
                                  MAY 31, 2016


Dear Shareholders:

Thank you for your investment in First Trust Energy Infrastructure Fund (the
"Fund").

First Trust Advisors L.P. ("First Trust") is pleased to provide you with the
semi-annual report which contains detailed information about your investment for
the six months ended May 31, 2016, including a market overview and a performance
analysis for the period. We encourage you to read this report and discuss it
with your financial advisor.

While markets were up and down during 2015, we believe there are three important
things to remember. First, the U.S. economy grew, despite the massive decline in
oil prices. Second, the tapering that began in 2014 by the Federal Reserve (the
"Fed") did not stop growth in the U.S. economy. Finally, the long-anticipated
rate hike by the Fed in December had little effect on the money supply, and the
stock market was not shocked by the hike. Early in 2016, many investors were
concerned that the volatility we saw in the market in 2015 would continue, and
it did. From December 31, 2015 through February 11, 2016, the S&P 500(R) Index
declined by 10.27%. Since then, the market has made a steady comeback, and as of
May 31, 2016, the S&P 500(R) Index was up 15.43%.

First Trust believes that having a long-term investment horizon and being
invested in quality products can help you reach your goals, regardless of how
the market behaves. We have always maintained perspective about the markets and
believe investors should as well. We will continue to strive to provide quality
investment opportunities each and every day, which has been one of the hallmarks
of our firm since its inception 25 years ago.

Thank you for giving First Trust the opportunity to be a part of your investment
plan. We value our relationship with you and will continue to focus on helping
investors like you reach your financial goals.

Sincerely,

/s/ James A. Bowen

James A. Bowen
Chairman of the Board of Trustees
Chief Executive Officer of First Trust Advisors L.P.


                                                                          Page 1





FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
"AT A GLANCE"
AS OF MAY 31, 2016 (UNAUDITED)

---------------------------------------------------------------------
FUND STATISTICS
---------------------------------------------------------------------
Symbol on New York Stock Exchange                                 FIF
Common Share Price                                             $16.56
Common Share Net Asset Value ("NAV")                           $18.74
Premium (Discount) to NAV                                      (11.63)%
Net Assets Applicable to Common Shares                   $328,878,818
Current Monthly Distribution per Common Share (1)             $0.1100
Current Annualized Distribution per Common Share              $1.3200
Current Distribution Rate on Closing Common Share Price (2)      7.97%
Current Distribution Rate on NAV (2)                             7.04%
---------------------------------------------------------------------

---------------------------------------------------------------------
           COMMON SHARE PRICE & NAV (WEEKLY CLOSING PRICE)
---------------------------------------------------------------------
            Common Share Price      NAV
5/15              $21.99          $24.84
                   21.00           23.81
                   20.60           23.33
                   21.01           23.57
6/15               20.51           23.54
                   19.99           22.96
                   19.81           23.03
                   19.25           22.72
                   18.50           21.50
7/15               18.89           22.31
                   17.78           20.98
                   18.64           21.88
                   17.80           20.46
8/15               17.97           20.91
                   17.25           19.70
                   16.61           19.17
                   17.28           19.62
9/15               16.60           19.11
                   16.79           19.45
                   18.98           20.64
                   18.96           20.63
                   18.20           19.84
10/15              18.20           19.94
                   17.03           19.21
                   15.62           18.13
                   16.09           18.35
11/15              16.16           18.19
                   14.24           16.05
                   13.05           15.22
                   13.30           15.35
                   14.87           16.67
12/15              14.59           16.54
                   13.42           15.66
                   12.06           14.53
                   12.84           15.23
1/16               13.97           16.12
                   13.74           16.09
                   12.92           14.89
                   13.62           15.85
2/16               14.26           16.17
                   15.33           16.88
                   15.45           17.04
                   15.76           17.52
3/16               15.01           17.16
                   15.14           17.08
                   15.04           17.18
                   15.33           17.48
                   16.29           18.24
4/16               16.75           18.48
                   16.12           18.15
                   16.30           18.34
                   16.88           18.56
                   16.53           18.64
5/16               16.56           18.74
---------------------------------------------------------------------




------------------------------------------------------------------------------------------------
PERFORMANCE
------------------------------------------------------------------------------------------------
                                                                              Average Annual
                                                                               Total Return
                                                                           ---------------------
                                           6 Months Ended   1 Year Ended    Inception (9/27/11)
                                              5/31/16         5/31/16           to 5/31/16
                                                                           
Fund Performance (3)
NAV                                            10.58%         -16.07%             10.76%
Market Value                                   10.00%         -16.23%              6.81%

Index Performance
Philadelphia Stock Exchange Utility Index      17.06%          13.34%             11.07%
Alerian MLP Total Return Index                  5.22%         -24.20%              3.45%
Blended Index (4)                              11.70%          -6.06%              7.78%
------------------------------------------------------------------------------------------------


------------------------------------------------------------
                                              % OF TOTAL
INDUSTRY CLASSIFICATION                       INVESTMENTS
------------------------------------------------------------
Pipelines                                        55.6%
Electric Power                                   34.9
Natural Gas Utility                               4.1
Propane                                           2.2
Coal                                              1.8
Gathering & Processing                            0.7
Other                                             0.7
------------------------------------------------------------
                                    Total       100.0%
                                                ======

------------------------------------------------------------
                                              % OF TOTAL
TOP 10 HOLDINGS                               INVESTMENTS
------------------------------------------------------------
Enbridge Energy Management, LLC                   9.1%
TransCanada Corp.                                 5.1
Kinder Morgan, Inc.                               5.1
Enterprise Products Partners, L.P.                4.5
Enbridge Income Fund Holdings, Inc. (CAD)         3.5
American Electric Power Co., Inc.                 3.2
NextEra Energy, Inc.                              3.2
Spectra Energy Corp.                              3.1
NextEra Energy Partners, L.P.                     2.9
Public Service Enterprise Group, Inc.             2.6
------------------------------------------------------------
                                    Total        42.3%
                                                ======


(1)   Most recent distribution paid or declared through 5/31/2016. Subject to
      change in the future.

(2)   Distribution rates are calculated by annualizing the most recent
      distribution paid or declared through the report date and then dividing by
      Common Share Price or NAV, as applicable, as of 5/31/2016. Subject to
      change in the future.

(3)   Total return is based on the combination of reinvested dividend, capital
      gain and return of capital distributions, if any, at prices obtained by
      the Dividend Reinvestment Plan and changes in NAV per share for NAV
      returns and changes in Common Share Price for market value returns. Total
      returns do not reflect sales load and are not annualized for periods less
      than one year. Past performance is not indicative of future results.

(4)   The blended index consists of the following: Philadelphia Stock Exchange
      Utility Index (50%) and Alerian MLP Total Return Index (50%).


Page 2





--------------------------------------------------------------------------------
PORTFOLIO COMMENTARY
--------------------------------------------------------------------------------

                  FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
                               SEMI-ANNUAL REPORT
                                  MAY 31, 2016

                                  SUB-ADVISOR


ENERGY INCOME PARTNERS, LLC

Energy Income Partners, LLC ("EIP" or the "Sub-Advisor"), located in Westport,
Connecticut, serves as the investment sub-advisor to the First Trust Energy
Infrastructure Fund (the "Fund"). EIP was founded in 2003 to provide
professional asset management services in the area of energy-related master
limited partnerships ("MLPs") and other high payout securities such as pipeline
companies, power utilities, yield corporations ("YieldCos")(1), and energy
infrastructure real estate investment trusts ("REITs"). EIP mainly focuses on
investments in energy-related infrastructure assets such as pipelines, power
transmission and distribution, petroleum storage and terminals that receive
fee-based or regulated income from their corporate and individual customers. EIP
manages or supervises approximately $4.7 billion of assets as of May 31, 2016.
Private funds advised by EIP include a partnership for U.S. high net worth
individuals and an open end registered investment company for institutions. EIP
also serves as an advisor to separately managed accounts for individuals and
institutions and provides its model portfolio to unified managed accounts.
Finally, EIP serves as a sub-advisor to three closed-end management investment
companies in addition to the Fund, an actively managed exchange-traded fund
("ETF"), a sleeve of an actively managed ETF and a sleeve of a series of a
variable insurance trust. EIP is a registered investment advisor with the
Securities and Exchange Commission.

                           PORTFOLIO MANAGEMENT TEAM

JAMES J. MURCHIE - PORTFOLIO MANAGER, FOUNDER, CHIEF EXECUTIVE OFFICER AND
   PRINCIPAL OF ENERGY INCOME PARTNERS, LLC

EVA PAO - CO-PORTFOLIO MANAGER, PRINCIPAL OF ENERGY INCOME PARTNERS, LLC

JOHN TYSSELAND - CO-PORTFOLIO MANAGER, PRINCIPAL OF ENERGY INCOME PARTNERS, LLC

                                   COMMENTARY

FIRST TRUST ENERGY INFRASTRUCTURE FUND

The investment objective of the Fund is to seek a high level of total return
with an emphasis on current distributions paid to shareholders. The Fund pursues
its objective by investing primarily in securities of companies engaged in the
energy infrastructure sector. These companies principally include
publicly-traded MLPs, MLP affiliates, YieldCos, pipeline companies, utilities
and other infrastructure-related companies that derive at least 50% of their
revenues from operating or providing services in support of infrastructure
assets such as pipelines, power transmission and petroleum and natural gas
storage in the petroleum, natural gas and power generation industries
(collectively, "Energy Infrastructure Companies"). Under normal market
conditions, the Fund invests at least 80% of its managed assets (total asset
value of the Fund minus the sum of the Fund's liabilities other than the
principal amount of borrowing) in securities of Energy Infrastructure Companies.
There can be no assurance that the Fund will achieve its investment objective.
The Fund may not be appropriate for all investors.

MARKET RECAP

As measured by the Alerian MLP Total Return Index ("AMZX") and the Philadelphia
Stock Exchange Utility Index ("UTY"), the total return for the six months ended
May 31, 2016 was 5.22% and 17.06%, respectively. These figures are according to
data collected from several sources, including Alerian Capital Management and
Bloomberg. While in the short term, share appreciation of Energy Infrastructure
Companies can be volatile, the Sub-Advisor believes that over the longer term,
such share appreciation will approximate growth in monthly cash distributions
and dividends per share. Over the last 10 years, growth in MLP distributions and
utility dividends has averaged 2.8% and 3.5%, respectively(2). Over the last 12
months, the cash distributions of MLPs represented by the Alerian MLP Total
Return Index decreased by about 12.9% and utilities increased by about 5.0%2.

PERFORMANCE ANALYSIS

On a net asset value ("NAV") basis, the Fund provided a total return(3) of
10.58%, including the reinvestment of distributions for the six months ended May
31, 2016. This compares, according to collected data, to an 11.70% return for a
blended index consisting of the UTY (50%) and the AMZX (50%) (the "Blended
Index"). Unlike the Fund, the Blended Index does not incur fees and expenses. On
a market value basis, the Fund had a total return, including the reinvestment of


(1)   YieldCos are publicly traded entities that own, operate and acquire
      contracted renewable and conventional generation and thermal and other
      infrastructure assets, which are generally not MLP-qualifying assets. Like
      MLPs, YieldCos generally seek to position themselves as vehicles for
      investors seeking stable and growing dividend income from a diversified
      portfolio of lower-risk, hiqh-quality assets.

(2)   Source: Alerian Capital Management, Bloomberg, EIP calculations.

(3)   Total return is based on the combination of reinvested dividends, capital
      gains and return of capital distributions, if any, at prices obtained by
      the Dividend Reinvestment Plan and changes in NAV per Common Share for NAV
      returns and changes in Common Share price for market value returns. Total
      returns do not reflect sales load and are not annualized for periods less
      than one year. Past performance is not indicative of future results.


                                                                          Page 3





--------------------------------------------------------------------------------
PORTFOLIO COMMENTARY (CONTINUED)
--------------------------------------------------------------------------------

                  FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
                               SEMI-ANNUAL REPORT
                                  MAY 31, 2016

distributions, of 10.00% for the same period. As of May 31, 2016, the Fund's
market price per share was $16.56, while the NAV per Common Share was $18.74, a
discount of 11.63%. As of November 30, 2015, the Fund's market price per share
was $16.16, while the NAV per Common Share was $18.19, a discount of 11.16%.

The Fund maintained its regular monthly Common Share distribution of $0.11 per
share for the six months ended May 31, 2016.

For the six months ended May 31, 2016, the Fund's NAV underperformed the 11.70%
return of the Blended Index by 112 basis points. As we have said numerous times,
we believe the MLP structure and a high payout ratio is only suitable for a
narrow set of long-lived assets that have stable non-cyclical cash flows, such
as regulated pipelines or other infrastructure assets that are legal or natural
monopolies. Nonetheless, the Fund underperformed the Blended Index during the
quarter as weakness in oil and gas prices continued to weigh on sentiment around
MLPs and other high payout energy infrastructure entities, including those that
can benefit from lower commodity prices. Falling security prices for companies
with high dividend payout ratios, in turn, hurts accretive growth from new
projects because more shares have to be issued to raise the same amount of
capital. These circumstances drove some managements to reduce dividends and
finance projects with internally generated cash flow to avoid selling equity at
dilutive prices. While this action preserves long-term value for shareholders,
companies in the portfolio that undertook such action were down and contributed
to the negative relative performance of the Fund.

An important factor that affected the return of the Fund was its use of
financial leverage through the use of a line of credit. The Fund has a committed
facility agreement with The Bank of Nova Scotia, with a maximum commitment
amount of $200,000,000. The Fund uses leverage because the Sub-Advisor believes
that, over time, leverage can enhance total return for common shareholders.
However, the use of leverage can also increase the volatility of the Fund's NAV
and therefore volatility of the Fund's Common Share price. For example, if the
prices of securities held by the Fund decline, the effect of changes in common
share NAV and common shareholder total return loss is magnified by the use of
leverage. Conversely, leverage may enhance Common Share returns during periods
when the prices of securities held by the Fund generally are rising. Unlike the
Fund, AMZX and UTY are not leveraged. Leverage had a positive impact on the
performance of the Fund over this reporting period.

MARKET AND FUND OUTLOOK

MLPs continue to play an integral role in the restructuring of more diversified
energy conglomerates. This restructuring includes the creation by these more
diversified conglomerates of subsidiaries with high dividend payout ratios that
own assets with stable cash flows such as pipelines, storage terminals and
electric power assets with long-term fixed-price contracts. The restructuring
can also include the divestiture by some of the parent companies of most or all
of their cyclical businesses, leaving the parent company looking very similar to
an old-fashioned pipeline utility with a large holding in a subsidiary MLP. We
believe these diversified energy conglomerates are restructuring so that their
regulated infrastructure assets that have predictable cash flows may be better
valued by the market. In our opinion, the result is a better financing tool to
raise capital for the new energy infrastructure projects. This phenomenon is
beginning to spread to the power utility industry, but instead of spinning out
MLPs, diversified power companies are spinning out regular "C" corporations with
higher dividend payout ratios (relative to earnings).

From January 1, 2016 through May 31, 2016, the MLP asset class experienced no
IPOs compared to 3 IPOs that raised $1.9 billion during the same time period in
2015. Reduced activity may be attributed to weak MLP equity markets as many
indexes were down sharply during the trailing 12 months. From January 1, 2016
through May 31, 2016, total MLP equity issuance was down 78% to $1.8 billion
compared to $8.4 billion over the same time period in 2015. Total MLP debt
transactions were down 81% to $5.4 billion from January 1, 2016 through May 31,
2016, which compares to $27.8 billion over the same time period in 2015,
according to Barclays MLP Weekly.

Capital expenditures for the 20 companies that comprise the UTY Index were $29.4
billion for the first quarter of 2016. The capital expenditure estimate for 2016
is $103.0 billion. This compares to $86.3 billion in 2014 and about $100.3
billion in 2015. (Source: Bloomberg) The expenditures are in response to needs
such as reliability, interconnection, modernization and growing demand. These
capital investments are supported, in part, by federal and state regulation,
which allows companies to recoup investments they have made in the rates they
charge their customers.

The Fund continues to seek to weight the portfolio toward Energy Infrastructure
Companies with mostly non-cyclical cash flows, investment-grade ratings,
conservative balance sheets, modest and/or flexible organic growth commitments
and available liquidity under revolving lines of credit. Since the Fund invests
in securities that tend to have high dividend payout ratios (relative to
earnings), in EIP's opinion, securities with unpredictable cyclical cash flows
make a poor fit within the portfolio. While there are some businesses within the
Fund's portfolio whose cash flows are cyclical, they are generally small and
analyzed in the context of each company's financial and operating leverage and
payout ratio.


Page 4





FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
PORTFOLIO OF INVESTMENTS
MAY 31, 2016 (UNAUDITED)



   SHARES                                      DESCRIPTION                                       VALUE
------------  -----------------------------------------------------------------------------  --------------
COMMON STOCKS - 83.8%

                                                                                       
              ELECTRIC UTILITIES - 20.1%
     191,400  American Electric Power Co., Inc.............................................   $  12,389,322
      53,000  Duke Energy Corp. (a)........................................................       4,146,190
     160,400  Emera, Inc. (CAD) (a)........................................................       5,652,254
     179,600  Eversource Energy (a)........................................................       9,921,104
     167,300  Exelon Corp. (a).............................................................       5,733,371
     146,800  Fortis, Inc. (CAD) (a).......................................................       4,592,013
      53,100  Hydro One Ltd. (CAD) (a) (b).................................................         989,640
      57,000  IDACORP, Inc. (a)............................................................       4,172,970
     102,300  NextEra Energy, Inc. (a).....................................................      12,288,276
      82,900  Southern (The) Co. (a).......................................................       4,098,576
      50,000  Xcel Energy, Inc. (a)........................................................       2,068,500
                                                                                             --------------
                                                                                                 66,052,216
                                                                                             --------------

              GAS UTILITIES - 3.8%
       5,800  Atmos Energy Corp. (a).......................................................         422,820
      84,810  Chesapeake Utilities Corp. (a)...............................................       4,892,689
     169,000  New Jersey Resources Corp. (a)...............................................       5,940,350
      17,700  ONE Gas, Inc. (a)............................................................       1,038,105
       8,718  UGI Corp.....................................................................         374,177
                                                                                             --------------
                                                                                                 12,668,141
                                                                                             --------------

              MULTI-UTILITIES - 18.3%
      60,000  Alliant Energy Corp. (a).....................................................       2,223,000
     142,800  ATCO, Ltd., Class I (CAD) (a)................................................       4,699,926
     160,000  Canadian Utilities Ltd., Class A (CAD) (a)...................................       4,422,923
     106,000  CMS Energy Corp..............................................................       4,432,920
     119,000  National Grid PLC, ADR (a)...................................................       8,783,390
     159,000  NiSource, Inc................................................................       3,793,740
     226,100  Public Service Enterprise Group, Inc. (a)....................................      10,117,975
      90,500  SCANA Corp. (a)..............................................................       6,326,855
      85,000  Sempra Energy (a)............................................................       9,105,200
     102,500  WEC Energy Group, Inc. (a)...................................................       6,164,350
                                                                                             --------------
                                                                                                 60,070,279
                                                                                             --------------

              OIL, GAS & CONSUMABLE FUELS - 41.6%
   1,623,538  Enbridge Energy Management, LLC (a) (c)......................................      35,490,541
     582,600  Enbridge Income Fund Holdings, Inc. (CAD) (a)................................      13,781,410
     215,169  Enbridge, Inc................................................................       8,583,091
     329,700  Inter Pipeline, Ltd. (CAD) (a)...............................................       6,700,351
     142,600  Keyera Corp. (CAD) (a).......................................................       4,254,022
   1,089,575  Kinder Morgan, Inc. (a)......................................................      19,699,516
     220,500  ONEOK, Inc...................................................................       9,536,625
     378,700  Spectra Energy Corp. (a).....................................................      12,065,382
     475,870  TransCanada Corp. (a)........................................................      19,729,570
     319,200  Williams (The) Cos., Inc. (a)................................................       7,073,472
                                                                                             --------------
                                                                                                136,913,980
                                                                                             --------------
              TOTAL COMMON STOCKS..........................................................     275,704,616
              (Cost $274,707,627)                                                            --------------



                        See Notes to Financial Statements                 Page 5





FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
PORTFOLIO OF INVESTMENTS (CONTINUED)
MAY 31, 2016 (UNAUDITED)



  SHARES/
   UNITS                                       DESCRIPTION                                       VALUE
------------  -----------------------------------------------------------------------------  --------------
MASTER LIMITED PARTNERSHIPS - 33.5%

                                                                                       
              CHEMICALS - 0.2%
      21,000  Westlake Chemical Partners, L.P. (a).........................................      $  482,790
                                                                                             --------------

              GAS UTILITIES - 2.5%
      87,752  AmeriGas Partners, L.P. (a)..................................................       4,026,062
     117,400  Suburban Propane Partners, L.P. (a)..........................................       4,057,344
                                                                                             --------------
                                                                                                  8,083,406
                                                                                             --------------

              INDEPENDENT POWER AND RENEWABLE ELECTRICITY PRODUCERS - 3.4%
     392,655  NextEra Energy Partners, L.P. (a)............................................      11,202,447
                                                                                             --------------

              OIL, GAS & CONSUMABLE FUELS - 27.4%
     138,000  Alliance Holdings GP, L.P. (a)...............................................       2,239,740
     329,430  Alliance Resource Partners, L.P. (a).........................................       4,826,150
      30,900  Buckeye Partners, L.P. (a)...................................................       2,222,328
     242,700  Columbia Pipeline Partners, L.P. (a).........................................       3,574,971
      36,100  Enbridge Energy Partners, L.P. (a)...........................................         784,814
     629,800  Enterprise Products Partners, L.P. (a).......................................      17,483,248
     108,700  EQT Midstream Partners, L.P. (a).............................................       8,192,719
     193,176  Holly Energy Partners, L.P. (a)..............................................       6,405,716
      58,600  Magellan Midstream Partners, L.P. (a)........................................       4,104,930
     183,472  NGL Energy Partners, L.P. (a)................................................       2,755,750
     143,000  ONEOK Partners, L.P. (a).....................................................       5,426,850
     425,986  Plains All American Pipeline, L.P. (a).......................................       9,853,056
     137,100  Spectra Energy Partners, L.P. (a)............................................       6,161,274
      77,328  Tallgrass Energy Partners, L.P. (a)..........................................       3,499,865
     115,195  TC Pipelines, L.P. (a).......................................................       6,362,220
      55,602  TransMontaigne Partners, L.P. (a)............................................       2,123,996
     131,438  Williams Partners, L.P. (a)..................................................       4,195,501
                                                                                             --------------
                                                                                                 90,213,128
                                                                                             --------------
              TOTAL MASTER LIMITED PARTNERSHIPS............................................     109,981,771
              (Cost $103,404,735)                                                            --------------

REAL ESTATE INVESTMENT TRUSTS - 0.9%

              REAL ESTATE INVESTMENT TRUSTS - 0.9%
     113,150  InfraREIT, Inc. (a)..........................................................       1,942,785
      43,801  CorEnergy Infrastructure Trust, Inc..........................................       1,053,414
                                                                                             --------------
              TOTAL REAL ESTATE INVESTMENT TRUSTS..........................................       2,996,199
              (Cost $3,990,526)                                                              --------------

              TOTAL INVESTMENTS - 118.2%...................................................     388,682,586
              (Cost $382,102,888) (d)


 NUMBER OF
 CONTRACTS                                     DESCRIPTION                                       VALUE
------------  -----------------------------------------------------------------------------  --------------
CALL OPTIONS WRITTEN - (0.7%)

              AMERICAN ELECTRIC POWER CO. INC. CALLS
       1,400  @ $67.50 due June 2016.......................................................         (14,000)
         500  @  65.00 due July 2016.......................................................         (58,000)
                                                                                             --------------
                                                                                                    (72,000)
                                                                                             --------------



Page 6                  See Notes to Financial Statements





FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
PORTFOLIO OF INVESTMENTS (CONTINUED)
MAY 31, 2016 (UNAUDITED)



 NUMBER OF
 CONTRACTS                                     DESCRIPTION                                       VALUE
------------  -----------------------------------------------------------------------------  --------------
CALL OPTIONS WRITTEN (CONTINUED)

                                                                                       
              CMS ENERGY CORP. CALL
       1,060  @ $40.00 due June 2016.......................................................  $     (185,500)
                                                                                             --------------

              ENBRIDGE INC. CALLS
         530  @  42.50 due June 2016.......................................................          (6,625)
         700  @  45.00 due June 2016 (e)...................................................          (3,500)
         300  @  42.50 due July 2016.......................................................         (14,250)
         600  @  45.00 due October 2016....................................................         (48,000)
                                                                                             --------------
                                                                                                    (72,375)
                                                                                             --------------

              EVERSOURCE ENERGY CALL
         900  @  60.00 due July 2016 (e)...................................................         (13,500)
                                                                                             --------------

              EXELON CORP. CALLS
         600  @  35.00 due June 2016.......................................................         (15,000)
         900  @  36.00 due June 2016.......................................................          (6,750)
                                                                                             --------------
                                                                                                    (21,750)
                                                                                             --------------

              KINDER MORGAN INC. CALLS
         900  @  20.00 due June 2016.......................................................          (3,600)
       8,300  @  20.00 due September 2016..................................................        (398,400)
                                                                                             --------------
                                                                                                   (402,000)
                                                                                             --------------

              NISOURCE INC. CALL
       1,590  @  24.00 due July 2016.......................................................        (112,890)
                                                                                             --------------

              ONEOK INC. CALLS
       1,000  @  45.00 due June 2016.......................................................         (70,000)
         920  @  47.50 due July 2016.......................................................         (73,600)
                                                                                             --------------
                                                                                                   (143,600)
                                                                                             --------------

              PLAINS ALL AMERICAN PIPELINE, L.P. CALLS
       2,700  @  27.00 due June 2016 (e)...................................................         (13,500)
         900  @  25.00 due July 2016.......................................................         (49,500)
                                                                                             --------------
                                                                                                    (63,000)
                                                                                             --------------

              PUBLIC SERVICE ENTERPRISE GROUP, INC. CALLS
         100  @  45.00 due June 2016.......................................................          (3,500)
       1,800  @  50.00 due June 2016 (e)...................................................          (5,400)
                                                                                             --------------
                                                                                                     (8,900)
                                                                                             --------------

              SEMPRA ENERGY CALL
         600  @ 105.00 due July 2016.......................................................        (192,000)
                                                                                             --------------

              SPECTRA ENERGY CORP. CALLS
         800  @  33.00 due June 2016.......................................................         (16,000)
       2,000  @  33.00 due July 2016.......................................................        (100,000)
         700  @  33.00 due September 2016..................................................         (67,900)
                                                                                             --------------
                                                                                                   (183,900)
                                                                                             --------------

              TRANSCANADA CORP. CALLS
         700  @  42.00 due July 2016 (e)...................................................         (57,937)
       1,300  @  40.00 due August 2016.....................................................        (319,800)
       1,400  @  45.00 due August 2016.....................................................         (52,500)
                                                                                             --------------
                                                                                                   (430,237)
                                                                                             --------------



                        See Notes to Financial Statements                 Page 7





FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
PORTFOLIO OF INVESTMENTS (CONTINUED)
MAY 31, 2016 (UNAUDITED)



 NUMBER OF
 CONTRACTS                                     DESCRIPTION                                       VALUE
------------  -----------------------------------------------------------------------------  --------------
CALL OPTIONS WRITTEN (CONTINUED)

                                                                                       
              WILLIAMS (THE) COS., INC. CALLS
       1,900  @ $20.00 due June 2016.......................................................  $     (461,700)
         800  @  23.00 due June 2016.......................................................         (52,800)
                                                                                             --------------
                                                                                                   (514,500)
                                                                                             --------------

              XCEL ENERGY INC. CALL
         300  @  45.00 due September 2016..................................................          (6,000)
                                                                                             --------------
              TOTAL CALL OPTIONS WRITTEN...................................................      (2,422,152)
              (Premiums received $2,018,077)                                                 --------------

              OUTSTANDING LOAN - (31.8%)...................................................    (104,500,000)

              NET OTHER ASSETS AND LIABILITIES - 14.3%.....................................      47,118,384
                                                                                             --------------
              NET ASSETS - 100.0%..........................................................  $  328,878,818
                                                                                             ==============


-----------------------------

(a)   All or a portion of this security serves as collateral on the outstanding
      loan.

(b)   This security is restricted in the U.S. and cannot be offered for public
      sale without first being registered under the Securities Act of 1933, as
      amended. This security is not restricted on the foreign exchange where it
      trades freely without any additional registration. As such, it does not
      require the additional disclosure required of restricted securities.

(c)   Non-income producing security that makes payment-in-kind ("PIK")
      distributions. For the fiscal year-to-date period (December 1, 2015
      through May 31, 2016), the Fund received 89,198 PIK shares of Enbridge
      Energy Management, LLC.

(d)   Aggregate cost for financial reporting purposes, which approximates the
      aggregate cost for federal income tax purposes. As of May 31, 2016, the
      aggregate gross unrealized appreciation for all securities in which there
      was an excess of value over tax cost was $34,811,084 and the aggregate
      gross unrealized depreciation for all securities in which there was an
      excess of tax cost over value was $28,231,386.

(e)   This security is fair valued by the First Trust Advisors L.P.'s Pricing
      Committee in accordance with procedures adopted by the Fund's Board of
      Trustees and in accordance with provisions of the Investment Company Act
      of 1940, as amended. At May 31, 2016, securities noted as such are valued
      at $(93,837) or (0.03)% of net assets.

ADR   American Depositary Receipt

CAD   Canadian Dollar - Security is denominated in Canadian Dollars and is
      translated into U.S. Dollars based upon the current exchange rate.

-----------------------------



INTEREST RATE SWAP AGREEMENTS:
                                                              NOTIONAL
   COUNTERPARTY       FLOATING RATE (1)   EXPIRATION DATE      AMOUNT       FIXED RATE(1)       VALUE
----------------------------------------------------------------------------------------------------------
                                                                             
Bank of Nova Scotia     1 month LIBOR        10/08/20       $  36,475,000      2.121%       $   (1,632,940)
Bank of Nova Scotia     1 month LIBOR        09/03/24          36,475,000      2.367%           (2,730,167)
                                                            -------------                   --------------
                                                            $  72,950,000                   $   (4,363,107)
                                                            =============                   ==============


(1)   The Fund pays the fixed rate and receives the floating rate. The floating
      rate on May 31, 2016 was 0.437% and 0.436%, respectively.


Page 8                  See Notes to Financial Statements





FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
PORTFOLIO OF INVESTMENTS (CONTINUED)
MAY 31, 2016 (UNAUDITED)

-----------------------------

VALUATION INPUTS

A summary of the inputs used to value the Fund's investments as of May 31, 2016
is as follows (see Note 3A - Portfolio Valuation in the Notes to Financial
Statements):



                                                    ASSETS TABLE
                                                                                         LEVEL 2          LEVEL 3
                                                        TOTAL           LEVEL 1        SIGNIFICANT      SIGNIFICANT
                                                       VALUE AT          QUOTED         OBSERVABLE      UNOBSERVABLE
                                                      5/31/2016          PRICES           INPUTS           INPUTS
                                                     ------------     ------------     ------------     ------------
                                                                                            
Common Stocks*.....................................  $275,704,616     $275,704,616     $         --     $         --
Master Limited Partnerships*.......................   109,981,771      109,981,771               --               --
Real Estate Investment Trusts*.....................     2,996,199        2,996,199               --               --
                                                     ------------     ------------     ------------     ------------
Total Investments..................................  $388,682,586     $388,682,586     $         --     $         --
                                                     ============     ============     ============     ============

                                                 LIABILITIES TABLE
                                                                                         LEVEL 2          LEVEL 3
                                                        TOTAL           LEVEL 1        SIGNIFICANT      SIGNIFICANT
                                                       VALUE AT          QUOTED         OBSERVABLE      UNOBSERVABLE
                                                      5/31/2016          PRICES           INPUTS           INPUTS
                                                     ------------     ------------     ------------     ------------
Call Options Written...............................  $ (2,422,152)    $ (2,328,315)    $    (93,837)    $         --
Interest Rate Swaps**..............................    (4,363,107)              --       (4,363,107)              --
                                                     ------------     ------------     ------------     ------------
Total..............................................  $ (6,785,259)    $ (2,328,315)    $ (4,456,944)    $         --
                                                     ============     ============     ============     ============



*   See Portfolio of Investments for industry breakout.
**  See Interest Rate Swap Agreements for contract detail.

All transfers in and out of the Levels during the period are assumed to be
transferred on the last day of the period at their current value. There were no
transfers between Levels at May 31, 2016.


                        See Notes to Financial Statements                 Page 9





FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
STATEMENT OF ASSETS AND LIABILITIES
MAY 31, 2016 (UNAUDITED)



                                                                                                  
ASSETS:
Investments, at value
     (Cost $382,102,888)........................................................................     $  388,682,586
Cash............................................................................................         42,418,406
Foreign currency (Cost $324,463)................................................................            322,583
Cash segregated as collateral for open swap contracts...........................................          6,562,424
Receivables:
   Investment securities sold...................................................................          1,859,901
   Dividends....................................................................................            911,108
   Interest.....................................................................................                278
Prepaid expenses................................................................................             13,410
                                                                                                     --------------
   Total Assets.................................................................................        440,770,696
                                                                                                     --------------
LIABILITIES:
Outstanding loan................................................................................        104,500,000
Options written, at value (Premiums received $2,018,077)........................................          2,422,152
Swap contracts, at value (Cost $569)............................................................          4,363,107
Payables:
   Investment advisory fees.....................................................................            361,735
   Interest and fees on loan....................................................................            134,554
   Audit and tax fees...........................................................................             30,260
   Investment securities purchased..............................................................             25,071
   Administrative fees..........................................................................             18,921
   Printing fees................................................................................             10,910
   Custodian fees...............................................................................              8,899
   Transfer agent fees..........................................................................              5,941
   Legal fees...................................................................................              5,916
   Trustees' fees and expenses..................................................................              3,641
   Financial reporting fees.....................................................................                771
                                                                                                     --------------
   Total Liabilities............................................................................        111,891,878
                                                                                                     --------------
NET ASSETS......................................................................................     $  328,878,818
                                                                                                     ==============
NET ASSETS CONSIST OF:
Paid-in capital.................................................................................     $  332,168,303
Par value.......................................................................................            175,502
Accumulated net investment income (loss)........................................................          8,935,016
Accumulated net realized gain (loss) on investments, written options, swap contracts and
   foreign currency transactions................................................................        (14,208,554)
Net unrealized appreciation (depreciation) on investments, written options, swap contracts
   and foreign currency translations............................................................          1,808,551
                                                                                                     --------------
NET ASSETS......................................................................................     $  328,878,818
                                                                                                     ==============
NET ASSET VALUE, per Common Share (par value $0.01 per Common Share)............................     $        18.74
                                                                                                     ==============
Number of Common Shares outstanding (unlimited number of Common Shares has been authorized).....         17,550,236
                                                                                                     ==============



Page 10                 See Notes to Financial Statements





FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
STATEMENT OF OPERATIONS
FOR THE SIX MONTHS ENDED MAY 31, 2016 (UNAUDITED)



                                                                                                  
INVESTMENT INCOME:
Dividends (net of foreign withholding tax of $243,358)..........................................     $    4,912,579
Interest........................................................................................                926
                                                                                                     --------------
   Total investment income......................................................................          4,913,505
                                                                                                     --------------
EXPENSES:
Investment advisory fees........................................................................          1,971,261
Interest and fees on loan.......................................................................            494,531
Administrative fees.............................................................................             97,690
Printing fees...................................................................................             86,998
Custodian fees..................................................................................             27,110
Audit and tax fees..............................................................................             25,150
Transfer agent fees.............................................................................             14,919
Trustees' fees and expenses.....................................................................              9,738
Legal fees......................................................................................              6,679
Financial reporting fees........................................................................              4,625
Other...........................................................................................             28,225
                                                                                                     --------------
   Total expenses...............................................................................          2,766,926
                                                                                                     --------------
NET INVESTMENT INCOME (LOSS)....................................................................          2,146,579
                                                                                                     --------------

NET REALIZED AND UNREALIZED GAIN (LOSS):
Net realized gain (loss) on:
   Investments..................................................................................         (6,566,224)
   Written options..............................................................................            574,578
   Swap contracts...............................................................................           (691,081)
   Foreign currency transactions................................................................            (90,909)
                                                                                                     --------------
Net realized gain (loss)........................................................................         (6,773,636)
                                                                                                     --------------
Net increase from payment by the sub-advisor....................................................              1,600
                                                                                                     --------------
Net change in unrealized appreciation (depreciation) on:
   Investments..................................................................................         34,249,971
   Written options..............................................................................         (1,051,146)
   Swap contracts...............................................................................         (1,425,995)
   Foreign currency translation.................................................................                (31)
                                                                                                     --------------
Net change in unrealized appreciation (depreciation)............................................         31,772,799
                                                                                                     --------------
NET REALIZED AND UNREALIZED GAIN (LOSS).........................................................         25,000,763
                                                                                                     --------------
NET INCREASE (DECREASE)  IN NET ASSETS RESULTING FROM OPERATIONS................................     $   27,147,342
                                                                                                     ==============



                        See Notes to Financial Statements                Page 11





FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
STATEMENTS OF CHANGES IN NET ASSETS



                                                                                           FOR THE
                                                                                          SIX MONTHS       FOR THE
                                                                                            ENDED            YEAR
                                                                                           5/312016         ENDED
                                                                                         (UNAUDITED)      11/30/2015
                                                                                        --------------  --------------
                                                                                                   
OPERATIONS:
Net investment income (loss).........................................................    $  2,146,579    $  5,952,005
Net realized gain (loss).............................................................      (6,773,636)     23,813,670
Net increase from payment by the sub-advisor.........................................           1,600              --
Net change in unrealized appreciation (depreciation).................................      31,772,799    (143,166,381)
                                                                                         ------------    ------------
Net increase (decrease) in net assets resulting from operations......................      27,147,342    (113,400,706)
                                                                                         ------------    ------------
DISTRIBUTIONS TO SHAREHOLDERS FROM:
Net investment income................................................................      (5,791,578)     (6,311,176)
Net realized gain....................................................................     (11,758,658)    (16,855,136)
                                                                                         ------------    ------------
Total distributions to shareholders..................................................     (17,550,236)    (23,166,312)
                                                                                         ------------    ------------
Total increase (decrease) in net assets..............................................       9,597,106    (136,567,018)

NET ASSETS:
Beginning of period..................................................................     319,281,712     455,848,730
                                                                                         ------------    ------------
End of period........................................................................    $328,878,818    $319,281,712
                                                                                         ============    ============
Accumulated net investment income (loss) at end of period............................    $  8,935,016    $ 12,580,015
                                                                                         ============    ============

COMMON SHARES:
Common Shares at end of period.......................................................      17,550,236      17,550,236
                                                                                         ============    ============




Page 12                 See Notes to Financial Statements





FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
STATEMENT OF CASH FLOWS
FOR THE SIX MONTHS ENDED MAY 31, 2016 (UNAUDITED)



                                                                                              
CASH FLOWS FROM OPERATING ACTIVITIES:
Net increase (decrease) in net assets resulting from operations .........   $   27,147,342
Adjustments to reconcile net increase (decrease) in net assets resulting
  from operations to net cash provided by operating activities:
   Purchases of investments..............................................     (135,871,599)
   Sales of investments..................................................      188,351,150
   Proceeds from written contracts.......................................        3,859,783
   Amount paid to close written options..................................       (1,525,180)
   Return of capital distributions received from investment in MLPs......        4,215,846
   Net realized gain/loss on investments and written options.............        5,991,646
   Net increase from payment by sub-advisor..............................           (1,600)
   Net change in unrealized appreciation/depreciation on investments and
     written options.....................................................      (33,198,825)
   Net change in unrealized appreciation/depreciation on swap contracts..        1,425,995
   Increase in cash segregated as collateral for open swap contracts.....       (1,208,919)

CHANGES IN ASSETS AND LIABILITIES:
   Increase in interest receivable.......................................             (223)
   Decrease in dividends receivable......................................          135,328
   Increase in prepaid expenses..........................................           (6,071)
   Increase in interest and fees on loan payable.........................           32,920
   Decrease in investment advisory fees payable..........................           (6,053)
   Decrease in audit and tax fees payable................................          (20,040)
   Increase in legal fees payable........................................            2,464
   Decrease in printing fees payable.....................................          (24,606)
   Decrease in administrative fees payable...............................             (722)
   Decrease in custodian fees payable....................................              (17)
   Increase in transfer agent fees payable...............................            3,052
   Increase in Trustees' fees and expenses payable.......................              299
   Decrease in other liabilities.........................................             (384)
                                                                            --------------
CASH USED IN OPERATING ACTIVITIES........................................                           $   59,301,586
                                                                                                    --------------

CASH FLOWS FROM FINANCING ACTIVITIES:
   Distributions to Common Shareholders from net investment income.......       (5,791,578)
   Distributions to Common Shareholders from net realized gain...........      (11,758,658)
   Proceeds from borrowing...............................................       10,000,000
   Repayment of borrowing................................................      (16,000,000)
                                                                            --------------
CASH USED IN FINANCING ACTIVITIES........................................                              (23,550,236)
                                                                                                    --------------
Increase in cash and foreign currency (a)................................                               35,751,350
Cash and foreign currency at beginning of period.........................                                6,989,639
                                                                                                    --------------
CASH AND FOREIGN CURRENCY AT END OF PERIOD...............................                           $   42,740,989
                                                                                                    ==============

SUPPLEMENTAL DISCLOSURE OF CASH FLOW INFORMATION:
Cash paid during the period for interest and fees........................                                $ 461,611
                                                                                                    ==============



-----------------------------

(a)   Includes net change in unrealized appreciation (depreciation) on foreign
      currency of $(31).


                        See Notes to Financial Statements                Page 13





FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
FINANCIAL HIGHLIGHTS
FOR A COMMON SHARE OUTSTANDING THROUGHOUT EACH PERIOD




                                               FOR THE
                                              SIX MONTHS                                                           FOR THE PERIOD
                                                ENDED                     YEAR ENDED NOVEMBER 30,                  9/27/2011 (a)
                                              5/30/2016    ------------------------------------------------------     THROUGH
                                             (UNAUDITED)       2015          2014          2013          2012        11/30/2011
                                             ------------  ------------  ------------  ------------  ------------  --------------
                                                                                                    
Net asset value, beginning of period........   $  18.19      $  25.97      $  22.30      $  22.74      $  21.38       $  19.10 (b)
                                               --------      --------      --------      --------      --------       --------
INCOME FROM INVESTMENT OPERATIONS:
Net investment income (loss)................       0.12          0.34          0.22          0.22          0.27           0.05
Net realized and unrealized gain (loss).....       1.43 (c)     (6.80)         6.20          3.47 (e)      2.38 (f)       2.30
                                               --------      --------      --------      --------      --------       --------
Total from investment operations............       1.55         (6.46)         6.42          3.69          2.65           2.35
                                               --------      --------      --------      --------      --------       --------
Common Shares offering costs charged to
   paid-in capital..........................         --            --            --            --          0.01          (0.04)
                                               --------      --------      --------      --------      --------       --------
Capital reduction from issuance of Common
   Shares related to over-allotment.........         --            --            --            --            --          (0.03)
                                               --------      --------      --------      --------      --------       --------
DISTRIBUTIONS PAID TO SHAREHOLDERS FROM:
Net investment income.......................      (0.33)        (0.36)        (0.45)        (0.83)        (1.01)            --
Net realized gain...........................      (0.67)        (0.96)        (2.21)        (3.25)        (0.29)            --
Return of capital...........................         --            --         (0.09)        (0.05)           --             --
                                               --------      --------      --------      --------      --------       --------
Total distributions to Common Shareholders..      (1.00)        (1.32)        (2.75)        (4.13)        (1.30)            --
                                               --------      --------      --------      --------      --------       --------
Net asset value, end of period..............   $  18.74      $  18.19      $  25.97      $  22.30      $  22.74       $  21.38
                                               ========      ========      ========      ========      ========       ========
Market value, end of period.................   $  16.56      $  16.16      $  23.00      $  21.71      $  21.34       $  19.82
                                               ========      ========      ========      ========      ========       ========
TOTAL RETURN BASED ON NET ASSET VALUE (d)...      10.58% (c)   (25.29)%       31.02%        17.76% (e)    13.08% (f)     11.94%
                                               ========      ========      ========      ========      ========       ========
TOTAL RETURN BASED ON MARKET VALUE (d)......      10.00%       (25.05)%       19.18%        22.11%        14.47%         (0.90)%
                                               ========      ========      ========      ========      ========       ========
-----------------------------

RATIOS TO AVERAGE NET ASSETS/SUPPLEMENTAL DATA:
Net assets, end of period (in 000's)........   $328,879      $319,282      $455,849      $391,336      $399,092       $375,232
Ratio of net expenses to average net
   assets...................................       1.89% (g)     1.83%         1.80%         1.84%         1.82%          1.66% (g)
Ratio of total expenses to average net
   assets excluding interest expense and
     fees on loan...........................       1.55% (g)     1.49%         1.53%         1.55%         1.52%          1.48% (g)
Ratio of net investment income (loss) to
   average net assets.......................       1.45% (g)     1.47%         0.88%         0.95%         1.21%          1.49% (g)
Portfolio turnover rate.....................         35%           42%           42%           54%           46%             8%
INDEBTEDNESS:
Total loan outstanding (in 000's)...........   $104,500      $110,500      $158,000      $145,900      $141,900       $102,000
Asset coverage per $1,000 of indebtedness
   (h)......................................   $  4,147      $  3,889      $  3,885      $  3,682      $  3,812       $  4,679



-----------------------------

(a)   Initial seed date of August 18, 2011. The Fund commenced operations on
      September 27, 2011.

(b)   Net of sales load of $0.90 per Common Share on initial offering.

(c)   The Fund received a reimbursement from the Sub-Advisor in the amount of
      $1,600 in connection with a trade error. The reimbursement from the
      Sub-Advisor represents less than $0.01 per share and had no effect on the
      Fund's total return.

(d)   Total return is based on the combination of reinvested dividend, capital
      gain and return of capital distributions, if any, at prices obtained by
      the Dividend Reinvestment Plan, and changes in net asset value per share
      for net asset value returns and changes in Common Share Price for market
      value returns. Total returns do not reflect sales load and are not
      annualized for periods of less than one year. Past performance is not
      indicative of future results.

(e)   The Fund received a reimbursement from the Sub-Advisor in the amount of
      $5,421 in connection with a trade error. The reimbursement from the
      Sub-Advisor represents less than $0.01 per common share and had no effect
      on the Fund's total return.

(f)   The Fund received a reimbursement from the Sub-Advisor in the amount of
      $104 in connection with a trade error. The reimbursement from the
      Sub-Advisor represents less than $0.01 per common share and had no effect
      on the Fund's total return.

(g)   Annualized.

(h)   Calculated by taking the Fund's total assets less the Fund's total
      liabilities, not including the loan outstanding, and dividing by the
      outstanding loan balance in 000's.


Page 14                 See Notes to Financial Statements





--------------------------------------------------------------------------------
NOTES TO FINANCIAL STATEMENTS
--------------------------------------------------------------------------------

                  FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
                            MAY 31, 2016 (UNAUDITED)


                                1. ORGANIZATION

First Trust Energy Infrastructure Fund (the "Fund") is a non-diversified,
closed-end management investment company organized as a Massachusetts business
trust on February 22, 2011 and is registered with the Securities and Exchange
Commission under the Investment Company Act of 1940, as amended (the "1940
Act"). The Fund trades under the ticker symbol FIF on the New York Stock
Exchange ("NYSE").

The Fund's investment objective is to seek a high level of total return with an
emphasis on current distributions paid to shareholders. The Fund seeks to
achieve its objective by investing primarily in securities of companies engaged
in the energy infrastructure sector. Energy infrastructure companies principally
include publicly-traded master limited partnerships and limited liability
companies taxed as partnerships ("MLPs"), MLP affiliates, YieldCos., pipeline
companies, utilities, and other companies that derive at least 50% of their
revenues from operating or providing services in support of infrastructure
assets such as pipelines, power transmission and petroleum and natural gas
storage in the petroleum, natural gas and power generation industries
(collectively, "Energy Infrastructure Companies"). For purposes of the Fund's
investment objective, total return includes capital appreciation of, and all
distributions received from, securities in which the Fund will invest, taking
into account the varying tax characteristics of such securities. There can be no
assurance that the Fund will achieve its investment objective. The Fund may not
be appropriate for all investors.

                         2. MANAGED DISTRIBUTION POLICY

The Board of Trustees of the Fund has approved a managed distribution policy for
the Fund (the "Plan") in reliance on exemptive relief received from the
Securities and Exchange Commission that permits the Fund to make periodic
distributions of long-term capital gains as frequently as monthly each tax year.
Under the Plan, the Fund currently intends to continue to pay a recurring
monthly distribution in the amount of $0.11 per Common Share that reflects the
distributable cash flow of the Fund. A portion of this monthly distribution may
include realized capital gains. This may result in a reduction of the long-term
capital gain distribution necessary at year end by distributing realized capital
gains throughout the year. The annual distribution rate is independent of the
Fund's performance during any particular period. Accordingly, you should not
draw any conclusions about the Fund's investment performance from the amount of
any distribution or from the terms of the Plan. The Board of Trustees may amend
or terminate the Plan at any time without prior notice to shareholders.

                       3. SIGNIFICANT ACCOUNTING POLICIES

The Fund, which is an investment company within the scope of Financial
Accounting Standards Board ("FASB") Accounting Standards Update 2013-08, follows
accounting and reporting guidance under FASB Accounting Standards Codification
Topic 946, "Financial Services-Investment Companies." The following is a summary
of significant accounting policies consistently followed by the Fund in the
preparation of its financial statements. The preparation of financial statements
in accordance with accounting principles generally accepted in the United States
of America ("U.S. GAAP") requires management to make estimates and assumptions
that affect the reported amounts and disclosures in the financial statements.
Actual results could differ from those estimates.

A. PORTFOLIO VALUATION

The net asset value ("NAV") of the Common Shares of the Fund is determined daily
as of the close of regular trading on the NYSE, normally 4:00 p.m. Eastern time,
on each day the NYSE is open for trading. If the NYSE closes early on a
valuation day, the NAV is determined as of that time. Foreign securities are
priced using data reflecting the earlier closing of the principal markets for
those securities. The Fund's NAV per Common Share is calculated by dividing the
value of all assets of the Fund (including accrued interest and dividends), less
all liabilities (including accrued expenses, dividends declared but unpaid and
any borrowings of the Fund) by the total number of Common Shares outstanding.

The Fund's investments are valued daily at market value or, in the absence of
market value with respect to any portfolio securities, at fair value. Market
value prices represent last sale or official closing prices from a national or
foreign exchange (i.e., a regulated market) and are primarily obtained from
third-party pricing services. Fair value prices represent any prices not
considered market value prices and are either obtained from a third-party
pricing service or are determined by the Pricing Committee of the Fund's
investment advisor, First Trust Advisors L.P. ("First Trust" or the "Advisor"),
in accordance with valuation procedures adopted by the Fund's Board of Trustees,
and in accordance with provisions of the 1940 Act. Investments valued by the
Advisor's Pricing Committee, if any, are footnoted as such in the footnotes to
the Portfolio of Investments. The Fund's investments are valued as follows:

      Common stocks, MLPs, real estate investment trusts ("REITs"), and other
      equity securities listed on any national or foreign exchange (excluding
      The Nasdaq Stock Market LLC ("Nasdaq") and the London Stock Exchange
      Alternative Investment Market ("AIM")) are valued at the last sale price
      on the exchange on which they are principally traded, or for Nasdaq and
      AIM securities, the official closing price. Securities traded on more than
      one securities exchange are valued at the last sale price or official
      closing price, as applicable, at the close of the securities exchange
      representing the principal market for such securities.

      Exchange-traded options contracts are valued at the closing price in the
      market where such contracts are principally traded. If no closing price is
      available, exchange-traded options contracts are fair valued at the mean
      of their most recent bid and asked price, if available, and otherwise at
      their closing bid price. Over-the-counter options contracts are fair
      valued at the mean of their most recent bid and asked price, if available,
      and otherwise their closing bid price.


                                                                         Page 15





--------------------------------------------------------------------------------
NOTES TO FINANCIAL STATEMENTS (CONTINUED)
--------------------------------------------------------------------------------

                  FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
                            MAY 31, 2016 (UNAUDITED)

      Securities traded in an over-the-counter market are fair valued at the
      mean of their most recent bid and asked price, if available, and otherwise
      at their closing bid price.

      Swaps are fair valued utilizing quotations provided by a third-party
      pricing service or, if the pricing service does not provide a value, by
      quotes provided by the selling dealer or financial institution.

Certain securities may not be able to be priced by pre-established pricing
methods. Such securities may be valued by the Fund's Board of Trustees or its
delegate, the Advisor's Pricing Committee, at fair value. These securities
generally include, but are not limited to, restricted securities (securities
which may not be publicly sold without registration under the Securities Act of
1933, as amended (the "1933 Act")), for which a pricing service is unable to
provide a market price; securities whose trading has been formally suspended; a
security whose market or fair value price is not available from a
pre-established pricing source; a security with respect to which an event has
occurred that is likely to materially affect the value of the security after the
market has closed but before the calculation of the Fund's NAV or make it
difficult or impossible to obtain a reliable market quotation; and a security
whose price, as provided by the pricing service, does not reflect the security's
fair value. As a general principle, the current fair value of a security would
appear to be the amount which the owner might reasonably expect to receive for
the security upon its current sale. When fair value prices are used, generally
they will differ from market quotations or official closing prices on the
applicable exchanges. A variety of factors may be considered in determining the
fair value of such securities, including, but not limited to, the following:

      1)    the type of security;

      2)    the size of the holding;

      3)    the initial cost of the security;

      4)    transactions in comparable securities;

      5)    price quotes from dealers and/or pricing services;

      6)    relationships among various securities;

      7)    information obtained by contacting the issuer, analysts, or the
            appropriate stock exchange;

      8)    an analysis of the issuer's financial statements; and

      9)    the existence of merger proposals or tender offers that might affect
            the value of the security.

If the securities in question are foreign securities, the following additional
information may be considered:

      1)    the value of similar foreign securities traded on other foreign
            markets;

      2)    ADR trading of similar securities;

      3)    closed-end fund trading of similar securities;

      4)    foreign currency exchange activity;

      5)    the trading prices of financial products that are tied to baskets of
            foreign securities;

      6)    factors relating to the event that precipitated the pricing problem;

      7)    whether the event is likely to recur; and

      8)    whether the effects of the event are isolated or whether they affect
            entire markets, countries or regions.

The Fund is subject to fair value accounting standards that define fair value,
establish the framework for measuring fair value and provide a three-level
hierarchy for fair valuation based upon the inputs to the valuation as of the
measurement date. The three levels of the fair value hierarchy are as follows:

      o     Level 1 - Level 1 inputs are quoted prices in active markets for
            identical investments. An active market is a market in which
            transactions for the investment occur with sufficient frequency and
            volume to provide pricing information on an ongoing basis.

      o     Level 2 - Level 2 inputs are observable inputs, either directly or
            indirectly, and include the following:

            o     Quoted prices for similar investments in active markets.

            o     Quoted prices for identical or similar investments in markets
                  that are non-active. A non-active market is a market where
                  there are few transactions for the investment, the prices are
                  not current, or price quotations vary substantially either
                  over time or among market makers, or in which little
                  information is released publicly.

            o     Inputs other than quoted prices that are observable for the
                  investment (for example, interest rates and yield curves
                  observable at commonly quoted intervals, volatilities,
                  prepayment speeds, loss severities, credit risks, and default
                  rates).

            o     Inputs that are derived principally from or corroborated by
                  observable market data by correlation or other means.

      o     Level 3 - Level 3 inputs are unobservable inputs. Unobservable
            inputs may reflect the reporting entity's own assumptions about the
            assumptions that market participants would use in pricing the
            investment.

The inputs or methodologies used for valuing investments are not necessarily an
indication of the risk associated with investing in those investments. A summary
of the inputs used to value the Fund's investments as of May 31, 2016, is
included with the Fund's Portfolio of Investments.


Page 16





--------------------------------------------------------------------------------
NOTES TO FINANCIAL STATEMENTS (CONTINUED)
--------------------------------------------------------------------------------

                  FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
                            MAY 31, 2016 (UNAUDITED)

B. OPTION CONTRACTS

The Fund is subject to equity price risk in the normal course of pursuing its
investment objective and may write (sell) options to hedge against changes in
the value of equities. Also, the Fund seeks to generate additional income, in
the form of premiums received, from writing (selling) the options. The Fund may
write (sell) covered call or put options ("options") on all or a portion of the
common stock and MLPs held in the Fund's portfolio as determined to be
appropriate by Energy Income Partners, LLC ("EIP" or the "Sub-Advisor"). The
number of options the Fund can write (sell) is limited by the amount of common
stock and MLPs the Fund holds in its portfolio. The Fund will not write (sell)
"naked" or uncovered options. Options are marked-to-market daily and their value
will be affected by changes in the value and dividend rates of the underlying
equity securities, changes in interest rates, changes in the actual or perceived
volatility of the securities markets and the underlying equity securities and
the remaining time to the options' expiration. The value of options may also be
adversely affected if the market for the options becomes less liquid or trading
volume diminishes.

Options the Fund writes (sells) will either be exercised, expire or be cancelled
pursuant to a closing transaction. If the price of the underlying equity
security exceeds the option's exercise price, it is likely that the option
holder will exercise the option. If an option written (sold) by the Fund is
exercised, the Fund would be obligated to deliver the underlying equity security
to the option holder upon payment of the strike price. In this case, the option
premium received by the Fund will be added to the amount realized on the sale of
the underlying security for purposes of determining gain or loss. If the price
of the underlying equity security is less than the option's strike price, the
option will likely expire without being exercised. The option premium received
by the Fund will, in this case, be treated as short-term capital gain on the
expiration date of the option. The Fund may also elect to close out its position
in an option prior to its expiration by purchasing an option of the same series
as the option written (sold) by the Fund. Gain or loss on options is presented
separately as "Net realized gain (loss) on written options" on the Statement of
Operations.

The options that the Fund writes (sells) give the option holder the right, but
not the obligation, to purchase a security from the Fund at the strike price on
or prior to the option's expiration date. The ability to successfully implement
the writing (selling) of covered call options depends on the ability of the
Sub-Advisor to predict pertinent market movements, which cannot be assured.
Thus, the use of options may require the Fund to sell portfolio securities at
inopportune times or for prices other than current market value, which may limit
the amount of appreciation the Fund can realize on an investment, or may cause
the Fund to hold a security that it might otherwise sell. As the writer (seller)
of a covered option, the Fund foregoes, during the option's life, the
opportunity to profit from increases in the market value of the security
covering the option above the sum of the premium and the strike price of the
option, but has retained the risk of loss should the price of the underlying
security decline. The writer (seller) of an option has no control over the time
when it may be required to fulfill its obligation as a writer (seller) of the
option. Once an option writer (seller) has received an exercise notice, it
cannot effect a closing purchase transaction in order to terminate its
obligation under the option and must deliver the underlying security to the
option holder at the exercise price.

Over-the-counter options have the risk of the potential inability of
counterparties to meet the terms of their contracts. The Fund's maximum equity
price risk for purchased options is limited to the premium initially paid. In
addition, certain risks may arise upon entering into option contracts including
the risk that an illiquid secondary market will limit the Fund's ability to
close out an option contract prior to the expiration date and that a change in
the value of the option contract may not correlate exactly with changes in the
value of the securities hedged.

C. SWAP AGREEMENTS

The Fund may enter into total return equity swap and interest rate swap
agreements. A swap is a financial instrument that typically involves the
exchange of cash flows between two parties ("Counterparties") on specified dates
(settlement dates) where the cash flows are based on agreed upon prices, rates,
etc. Payment received or made by the fund for interest rate swaps are recorded
on the Statement of Operations as "Net realized gain (loss) on swap contracts."
When an interest rate swap is terminated, the Fund will record a realized gain
or loss equal to the difference between the proceeds from (or cost of) the
closing transaction and the Fund's basis in the contract, if any. Generally, the
basis of the contracts, if any, is the premium received or paid. Swap agreements
are individually negotiated and involve the risk of the potential inability of
the Counterparties to meet the terms of the agreement. In connection with these
agreements, cash and securities may be identified as collateral in accordance
with the terms of the respective swap agreements to provide assets of value and
recourse in the event of default under the swap agreement or
bankruptcy/insolvency of a party to the swap agreement. In the event of a
default by a Counterparty, the Fund will seek withdrawal of the collateral and
may incur certain costs exercising its rights with respect to the collateral. If
a Counterparty becomes bankrupt or otherwise fails to perform its obligations
due to financial difficulties, the Fund may experience significant delays in
obtaining any recovery in a bankruptcy or other reorganization proceeding. The
Fund may obtain only limited recovery or may obtain no recovery in such
circumstances.

Swap agreements may increase or decrease the overall volatility of the
investments of the Fund. The performance of swap agreements may be affected by
changes in the specific interest rate, security, currency, or other factors that
determine the amounts of payments due to and from the Fund. The Fund's maximum
equity price risk to meet its future payments under swap agreements outstanding
at May 31, 2016 is equal to the total notional amount as shown on the Portfolio
of Investments. The notional amount represents the U.S. dollar value of the
contract as of the day of the opening transaction or contract reset. When the
Fund enters into a swap agreement, any premium paid is included in "Swap
contracts, at value" on the Statement of Assets and Liabilities.


                                                                         Page 17





--------------------------------------------------------------------------------
NOTES TO FINANCIAL STATEMENTS (CONTINUED)
--------------------------------------------------------------------------------

                  FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
                            MAY 31, 2016 (UNAUDITED)

The Fund held interest rate swap agreements at May 31, 2016 to hedge against
changes in borrowing rates under the Fund's committed facility agreement. An
interest rate swap agreement involves the Fund's agreement to exchange a stream
of interest payments for another party's stream of cash flows. Interest rate
swaps do not involve the delivery of securities or other underlying assets or
principal. Accordingly, the risk of loss with respect to interest rate swaps is
limited to the net amount of interest payments that the Fund is contractually
obligated to make. Cash segregated as collateral for open swap contracts in the
amount of $6,562,424 is shown on the Statement of Assets and Liabilities.

D. SECURITIES TRANSACTIONS AND INVESTMENT INCOME

Securities transactions are recorded as of the trade date. Realized gains and
losses from securities transactions are recorded on the identified cost basis.
Dividend income is recorded on the ex-dividend date. Interest income is recorded
daily on the accrual basis. The Fund will rely to some extent on information
provided by MLPs, which is not necessarily timely, to estimate taxable income
allocable to the MLP units held in the Fund's portfolio.

Distributions received from the Fund's investments in MLPs generally are
comprised of return of capital and investment income. The Fund records estimated
return of capital and investment income based on historical information
available from each MLP. These estimates may subsequently be revised based on
information received from the MLPs after their tax reporting periods are
concluded. For the six months ended May 31, 2016, distributions of $4,215,846
received from MLPs have been reclassified as return of capital. The cost basis
of applicable MLPs has been reduced accordingly.

The Fund holds shares of REITs. Distributions from such investments may be
comprised of return of capital, capital gains and income. The actual character
of amounts received during the year is not known until after the REIT's fiscal
year end. The Fund records the character of distributions received from REITs
during the year based on estimates available. The characterization of
distributions received by the Fund may be subsequently revised based on
information received from the REITs after their tax reporting periods conclude.

E. DIVIDENDS AND DISTRIBUTIONS TO SHAREHOLDERS

The Fund will distribute to holders of its Common Shares monthly dividends of
all or a portion of its net income after the payment of interest in connection
with leverage, if any. Distributions of any long-term capital gains earned by
the Fund are distributed at least annually. Distributions will automatically be
reinvested into additional Common Shares pursuant to the Fund's Dividend
Reinvestment Plan unless cash distributions are elected by the shareholder.

Distributions from income and capital gains are determined in accordance with
federal income tax regulations, which may differ from U.S. GAAP. Certain capital
accounts in the financial statements are periodically adjusted for permanent
differences in order to reflect their tax character. These permanent differences
are primarily due to the varying treatment of income and gain/loss on portfolio
securities held by the Fund and have no impact on net assets or NAV per Common
Share. Temporary differences, which arise from recognizing certain items of
income, expense and gain/loss in different periods for financial statement and
tax purposes, will reverse at some point in the future.

The tax character of distributions paid during the fiscal year ended November
30, 2015 was as follows:

Distributions paid from:                                2015
Ordinary income...................................  $   9,729,196
Capital gains.....................................     13,437,116
Return of capital.................................             --


Page 18





--------------------------------------------------------------------------------
NOTES TO FINANCIAL STATEMENTS (CONTINUED)
--------------------------------------------------------------------------------

                  FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
                            MAY 31, 2016 (UNAUDITED)

As of November 30, 2015, the components of distributable earnings and net assets
on a tax basis were as follows:

Undistributed ordinary income.....................  $          --
Undistributed capital gains.......................      5,326,969
                                                    -------------
Total undistributed earnings......................      5,326,969
Accumulated capital and other losses..............             --
Net unrealized appreciation (depreciation)........    (18,382,135)
                                                    -------------
Total accumulated earnings (losses)...............    (13,055,166)
Other.............................................         (6,927)
Paid-in capital...................................    332,343,805
                                                    -------------
Net assets........................................  $ 319,281,712
                                                    =============

F. INCOME TAXES

The Fund intends to continue to qualify as a regulated investment company by
complying with the requirements under Subchapter M of the Internal Revenue Code
of 1986, as amended, which includes distributing substantially all of its net
investment income and net realized gains to shareholders. Accordingly, no
provision has been made for federal and state income taxes. However, due to the
timing and amount of distributions, the Fund may be subject to an excise tax of
4% of the amount by which approximately 98% of the Fund's taxable income exceeds
the distributions from such taxable income for the calendar year.

The Fund intends to utilize provisions of the federal income tax laws, which
allow it to carry a realized capital loss forward indefinitely following the
year of the loss and offset such loss against any future realized capital gains.
The Fund is subject to certain limitations under U.S. tax rules on the use of
capital loss carryforwards and net unrealized built-in losses. These limitations
apply when there has been a 50% change in ownership. At November 30, 2015, the
Fund had no non-expiring capital loss carryforwards for federal income tax
purposes.

Certain losses realized during the current fiscal year may be deferred and
treated as occurring on the first day of the following fiscal year for federal
income tax purposes. For the fiscal year ended November 30, 2015, the Fund had
no qualified late year losses.

The Fund is subject to accounting standards that establish a minimum threshold
for recognizing, and a system for measuring, the benefits of a tax position
taken or expected to be taken in a tax return. Taxable years ending 2012, 2013,
2014 and 2015 remain open to federal and state audit. As of May 31, 2016,
management has evaluated the application of these standards to the Fund and has
determined that no provision for income tax is required in the Fund's financial
statements for uncertain tax positions.

G. EXPENSES

The Fund will pay all expenses directly related to its operations.

H. FOREIGN CURRENCY

The books and records of the Fund are maintained in U.S. dollars. Foreign
currencies, investments and other assets and liabilities are translated into
U.S. dollars at the exchange rates prevailing at the end of the period.
Purchases and sales of investments and items of income and expense are
translated on the respective dates of such transactions. Unrealized gains and
losses on assets and liabilities, other than investments in securities, which
result from changes in foreign currency exchange rates have been included in
"Net change in unrealized appreciation (depreciation) on foreign currency
translation" on the Statement of Operations. Unrealized gains and losses on
investments in securities which result from changes in foreign exchange rates
are included with fluctuations arising from changes in market price and are
shown in "Net change in unrealized appreciation (depreciation) on investments"
on the Statement of Operations. Net realized foreign currency gains and losses
include the effect of changes in exchange rates between trade date and
settlement date on investment security transactions, foreign currency
transactions and interest and dividends received and are shown in "Net realized
gain (loss) on foreign currency transactions" on the Statement of Operations.
The portion of foreign currency gains and losses related to fluctuation in
exchange rates between the initial purchase settlement date and subsequent sale
trade date is included in "Net realized gain (loss) on investments" on the
Statement of Operations.

I. OFFSETTING ON THE STATEMENT OF ASSETS AND LIABILITIES

Offsetting Assets and Liabilities require entities to disclose both gross and
net information about instruments and transactions eligible for offset on the
Statements of Assets and Liabilities, and disclose instruments and transactions
subject to master netting or similar agreements. These disclosure requirements
are intended to help investors and other financial statement users better assess
the effect or potential effect of offsetting arrangements on a fund's financial
position. The transactions subject to offsetting disclosures are derivative
instruments, repurchase agreements and reverse repurchase agreements, and
securities borrowing and securities lending transactions.


                                                                         Page 19





--------------------------------------------------------------------------------
NOTES TO FINANCIAL STATEMENTS (CONTINUED)
--------------------------------------------------------------------------------

                  FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
                            MAY 31, 2016 (UNAUDITED)

For financial reporting purposes, the Fund does not offset financial assets and
financial liabilities that are subject to master netting arrangements ("MNAs")
or similar agreements on the Statement of Assets and Liabilities. MNAs provide
the right, in the event of default (including bankruptcy and insolvency), for
the non-defaulting counterparty to liquidate the collateral and calculate the
net exposure to the defaulting party or request additional collateral.

At May 31, 2016, derivative assets and liabilities (by type) on a gross basis
are as follows:



                                                                                           GROSS AMOUNTS NOT OFFSET
                                                                                             IN THE STATEMENT OF
                                                                      NET AMOUNTS OF        ASSETS AND LIABILITIES
                                   GROSS         GROSS AMOUNTS     LIABILITIES PRESENTED  --------------------------
                                AMOUNTS OF       OFFSET IN THE       IN THE STATEMENT                      CASH
                                RECOGNIZED    STATEMENT OF ASSETS      OF ASSETS AND       FINANCIAL   SEGREGATED AS     NET
                                LIABILITIES     AND LIABILITIES         LIABILITIES       INSTRUMENTS   COLLATERAL     AMOUNT
-------------------------------------------------------------------------------------------------------------------------------
                                                                                                      
Interest Rate Swap Contracts$   (4,363,107)          $ --              $ (4,363,107)         $ --      $ 4,363,107      $ --



4. INVESTMENT ADVISORY FEE, AFFILIATED TRANSACTIONS AND OTHER FEE ARRANGEMENTS

First Trust, the investment advisor to the Fund, is a limited partnership with
one limited partner, Grace Partners of DuPage L.P., and one general partner, The
Charger Corporation. The Charger Corporation is an Illinois corporation
controlled by James A. Bowen, Chief Executive Officer of First Trust. First
Trust is responsible for the ongoing monitoring of the Fund's investment
portfolio, managing the Fund's business affairs and providing certain
administrative services necessary for the management of the Fund. For these
investment management services, First Trust is entitled to a monthly fee
calculated at an annual rate of 1.00% of the Fund's Managed Assets (the average
daily total asset value of the Fund minus the sum of the Fund's liabilities
other than the principal amount of borrowings). First Trust also provides fund
reporting services to the Fund for a flat annual fee in the amount of $9,250.

EIP serves as the Fund's sub-advisor and manages the Fund's portfolio subject to
First Trust's supervision. The Sub-Advisor receives a monthly sub-advisory fee
calculated at an annual rate of 0.50% of the Fund's Managed Assets that is paid
by First Trust out of its investment advisory fee.

During the six months ended May 31, 2016, the Fund received a payment from the
sub-advisor of $1,600 in connection with a trade error.

First Trust Capital Partners, LLC, an affiliate of First Trust, owns, through a
wholly-owned subsidiary, a 15% ownership interest in each of EIP and EIP
Partners, LLC, an affiliate of the EIP.

BNY Mellon Investment Servicing (US) Inc. ("BNYM IS") serves as the Fund's
administrator, fund accountant and transfer agent in accordance with certain fee
arrangements. As administrator and fund accountant, BNYM IS is responsible for
providing certain administrative and accounting services to the Fund, including
maintaining the Fund's books of account, records of the Fund's securities
transactions, and certain other books and records. As transfer agent, BNYM IS is
responsible for maintaining shareholder records for the Fund. The Bank of New
York Mellon ("BNYM") serves as the Fund's custodian in accordance with certain
fee arrangements. As custodian, BNYM is responsible for custody of the Fund's
assets. BNYM IS and BNYM are subsidiaries of The Bank of New York Mellon
Corporation, a financial holding company.

Each Trustee who is not an officer or employee of First Trust, any sub-advisor
or any of their affiliates ("Independent Trustees") is paid a fixed annual
retainer that is allocated equally among each fund in the First Trust Fund
Complex. Prior to January 1, 2016, the fixed annual retainer was allocated pro
rata based on each fund's net assets. Each Independent Trustee is also paid an
annual per fund fee that varies based on whether the fund is a closed-end or
other actively managed fund, or is an index fund.

Additionally, the Lead Independent Trustee and the Chairmen of the Audit
Committee, Nominating and Governance Committee and Valuation Committee are paid
annual fees to serve in such capacities, with such compensation allocated pro
rata among each fund in the First Trust Fund Complex based on net assets.
Trustees are reimbursed for travel and out-of-pocket expenses in connection with
all meetings. The Lead Independent Trustee and Committee Chairmen rotate every
three years. The officers and "Interested" Trustee receive no compensation from
the Fund for acting in such capacities.

                      5. PURCHASES AND SALES OF SECURITIES

Cost of purchases and proceeds from sales of investments, excluding short-term
investments, for the six months ended May 31, 2016, were $135,896,670 and
$190,733,271, respectively.


Page 20





--------------------------------------------------------------------------------
NOTES TO FINANCIAL STATEMENTS (CONTINUED)
--------------------------------------------------------------------------------

                  FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
                            MAY 31, 2016 (UNAUDITED)


                           6. DERIVATIVE TRANSACTIONS

For the fiscal year-to-date period (December 1, 2015 through May 31, 2016),
written option activity for the Fund was as follows:

                                                  NUMBER OF
WRITTEN OPTIONS                                   CONTRACTS        PREMIUMS
-------------------------------------------------------------------------------
Options outstanding at November 30, 2015...           25,130     $  1,352,376
Options Written............................           71,780        3,859,783
Options Expired............................          (33,136)      (1,618,003)
Options Exercised..........................          (19,794)      (1,094,325)
Options Closed.............................           (7,780)        (481,754)
                                                  ----------     ------------
Options outstanding at May 31, 2016........           36,200     $  2,018,077
                                                  ==========     ============

The following table presents the types of derivatives held by the Fund at May
31, 2016, the primary underlying risk exposure and the location of these
instruments as presented on the Statement of Assets and Liabilities.



                                                   ASSET DERIVATIVES                      LIABILITY DERIVATIVES
                                        ---------------------------------------  ---------------------------------------
    DERIVATIVE             RISK          STATEMENT OF ASSETS AND                  STATEMENT OF ASSETS AND
   INSTRUMENTS           EXPOSURE         LIABILITIES LOCATION        VALUE        LIABILITIES LOCATION        VALUE
------------------  ------------------  -------------------------  ------------  -------------------------  ------------
                                                                     
Written Options     Equity Risk                    --                   --       Options written, at value  $  2,422,152

Interest Rate Swap
Agreement           Interest Rate Risk             --                   --       Swap contracts, at value      4,363,107


The following table presents the amount of net realized gain (loss) and change
in net unrealized appreciation (depreciation) recognized for the six months
ended May 31, 2016, on derivative instruments, as well as the primary underlying
risk exposure associated with each instrument.

STATEMENT OF OPERATIONS LOCATION
------------------------------------------------------------------------
EQUITY RISK EXPOSURE
Net realized gain (loss) on written options                  $   574,578

Net change in unrealized appreciation
   (depreciation) on written options                          (1,051,146)

INTEREST RATE RISK EXPOSURE
Net realized gain (loss) on swap contracts                      (691,081)

Net change in unrealized appreciation
   (depreciation) on swap contracts                           (1,425,995)

The average volume of interest rate swaps was $72,950,000 for the six months
ended May 31, 2016.

The Fund does not have the right to offset financial assets and financial
liabilities related to option contracts on the Statement of Assets and
Liabilities.

                                 7. BORROWINGS

During the period covered by this report, the Fund maintained a committed
facility agreement with The Bank of Nova Scotia ("Scotia") with a maximum
commitment amount of $200,000,000 and a borrowing rate equal to the 1-month
LIBOR plus 70 basis points. In addition, under the facility, the Fund pays a
commitment fee of 0.15% when the undrawn amount of such facility is greater than
50% of the maximum commitment. The average amount outstanding under the facility
for the six months ended May 31, 2016 was $100,926,230, with a weighted average
interest rate of 1.20%. As of May 31, 2016, the Fund had outstanding borrowings
of $104,500,000 under this committed facility agreement. The high and low annual
interest rates for the six months ended May 31, 2016, were 1.32% and 0.94%,
respectively. The interest rate at May 31, 2016 was 1.32%.

                               8. INDEMNIFICATION

The Fund has a variety of indemnification obligations under contracts with its
service providers. The Fund's maximum exposure under these arrangements is
unknown. However, the Fund has not had prior claims or losses pursuant to these
contracts and expects the risk of loss to be remote.


                                                                         Page 21





--------------------------------------------------------------------------------
NOTES TO FINANCIAL STATEMENTS (CONTINUED)
--------------------------------------------------------------------------------

                  FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
                            MAY 31, 2016 (UNAUDITED)


                         9. INDUSTRY CONCENTRATION RISK

The Fund invests at least 80% of its Managed Assets in securities issued by
Energy Infrastructure Companies. Given this industry concentration, the Fund is
more susceptible to adverse economic or regulatory occurrences affecting that
industry than an investment company that is not concentrated in a single
industry. Energy Infrastructure Company issuers may be subject to a variety of
factors that may adversely affect their business or operations, including high
interest costs in connection with capital construction programs, high leverage
costs associated with environmental and other regulations, the effects of
economic slowdown, surplus capacity, increased competition from other providers
of services, uncertainties concerning the availability of fuel at reasonable
prices, the effects of energy conservation policies and other factors.

               10. NON-FUNDAMENTAL CHANGE IN INVESTMENT STRATEGY

On February 1, 2016, the Board of Trustees for the Fund approved the following
non-fundamental change to Fund's investment strategy:

      The Fund's covered call limit was increased from 30% to 35%. Accordingly,
      effective on or about April 11, 2016, the Fund may write (or sell) covered
      call options on up to 35% of its Managed Assets.

The Fund's investment objective did not change.

                              11. SUBSEQUENT EVENT

Management has evaluated the impact of all subsequent events to the Fund through
the date the financial statements were issued and has determined that there was
the following subsequent event:

On June 20, 2016, the Fund declared a distribution of $0.11 per share to Common
Shareholders of record on July 6, 2016, payable July 15, 2016.


Page 22





--------------------------------------------------------------------------------
ADDITIONAL INFORMATION
--------------------------------------------------------------------------------

                  FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
                            MAY 31, 2016 (UNAUDITED)


                           DIVIDEND REINVESTMENT PLAN

If your Common Shares are registered directly with the Fund or if you hold your
Common Shares with a brokerage firm that participates in the Fund's Dividend
Reinvestment Plan (the "Plan"), unless you elect, by written notice to the Fund,
to receive cash distributions, all dividends, including any capital gain
distributions, on your Common Shares will be automatically reinvested by BNY
Mellon Investment Servicing (US) Inc. (the "Plan Agent"), in additional Common
Shares under the Plan. If you elect to receive cash distributions, you will
receive all distributions in cash paid by check mailed directly to you by the
Plan Agent, as the dividend paying agent.

If you decide to participate in the Plan, the number of Common Shares you will
receive will be determined as follows:

      (1)   If Common Shares are trading at or above net asset value ("NAV") at
            the time of valuation, the Fund will issue new shares at a price
            equal to the greater of (i) NAV per Common Share on that date or
            (ii) 95% of the market price on that date.

      (2)   If Common Shares are trading below NAV at the time of valuation, the
            Plan Agent will receive the dividend or distribution in cash and
            will purchase Common Shares in the open market, on the NYSE or
            elsewhere, for the participants' accounts. It is possible that the
            market price for the Common Shares may increase before the Plan
            Agent has completed its purchases. Therefore, the average purchase
            price per share paid by the Plan Agent may exceed the market price
            at the time of valuation, resulting in the purchase of fewer shares
            than if the dividend or distribution had been paid in Common Shares
            issued by the Fund. The Plan Agent will use all dividends and
            distributions received in cash to purchase Common Shares in the open
            market within 30 days of the valuation date except where temporary
            curtailment or suspension of purchases is necessary to comply with
            federal securities laws. Interest will not be paid on any uninvested
            cash payments.

You may elect to opt-out of or withdraw from the Plan at any time by giving
written notice to the Plan Agent, or by telephone at (866) 340-1104, in
accordance with such reasonable requirements as the Plan Agent and the Fund may
agree upon. If you withdraw or the Plan is terminated, you will receive a
certificate for each whole share in your account under the Plan, and you will
receive a cash payment for any fraction of a share in your account. If you wish,
the Plan Agent will sell your shares and send you the proceeds, minus brokerage
commissions.

The Plan Agent maintains all Common Shareholders' accounts in the Plan and gives
written confirmation of all transactions in the accounts, including information
you may need for tax records. Common Shares in your account will be held by the
Plan Agent in non-certificated form. The Plan Agent will forward to each
participant any proxy solicitation material and will vote any shares so held
only in accordance with proxies returned to the Fund. Any proxy you receive will
include all Common Shares you have received under the Plan.

There is no brokerage charge for reinvestment of your dividends or distributions
in Common Shares. However, all participants will pay a pro rata share of
brokerage commissions incurred by the Plan Agent when it makes open market
purchases.

Automatically reinvesting dividends and distributions does not mean that you do
not have to pay income taxes due upon receiving dividends and distributions.
Capital gains and income are realized although cash is not received by you.
Consult your financial advisor for more information.

If you hold your Common Shares with a brokerage firm that does not participate
in the Plan, you will not be able to participate in the Plan and any dividend
reinvestment may be effected on different terms than those described above.

The Fund reserves the right to amend or terminate the Plan if in the judgment of
the Board of Trustees the change is warranted. There is no direct service charge
to participants in the Plan; however, the Fund reserves the right to amend the
Plan to include a service charge payable by the participants. Additional
information about the Plan may be obtained by writing BNY Mellon Investment
Servicing (US) Inc., 301 Bellevue Parkway, Wilmington, Delaware 19809.

--------------------------------------------------------------------------------

                      PROXY VOTING POLICIES AND PROCEDURES

A description of the policies and procedures that the Fund uses to determine how
to vote proxies and information on how the Fund voted proxies relating to
portfolio investments during the most recent 12-month period ended June 30 is
available (1) without charge, upon request, by calling (800) 988-5891; (2) on
the Fund's website located at http://www.ftportfolios.com; and (3) on the
Securities and Exchange Commission's ("SEC") website located at
http://www.sec.gov.


                                                                         Page 23





--------------------------------------------------------------------------------
ADDITIONAL INFORMATION (CONTINUED)
--------------------------------------------------------------------------------

                  FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
                            MAY 31, 2016 (UNAUDITED)


                               PORTFOLIO HOLDINGS

The Fund files its complete schedule of portfolio holdings with the SEC for the
first and third quarters of each fiscal year on Form N-Q. The Fund's Form N-Qs
are available (1) by calling (800) 988-5891; (2) on the Fund's website located
at http://www.ftportfolios.com; (3) on the SEC's website at http://www.sec.gov;
and (4) for review and copying at the SEC's Public Reference Room ("PRR") in
Washington, DC. Information regarding the operation of the PRR may be obtained
by calling (800) SEC-0330.

                    SUBMISSION OF MATTERS TO A VOTE OF SHAREHOLDERS

The Joint Annual Meeting of Shareholders of the Common Shares of Macquarie/First
Trust Global Infrastructure/Utilities Dividend & Income Fund, First Trust Energy
Income and Growth Fund, First Trust Enhanced Equity Income Fund, First
Trust/Aberdeen Global Opportunity Income Fund, First Trust Mortgage Income Fund,
First Trust Strategic High Income Fund II, First Trust/Aberdeen Emerging
Opportunity Fund, First Trust Specialty Finance and Financial Opportunities
Fund, First Trust High Income Long/Short Fund, First Trust Energy Infrastructure
Fund, First Trust MLP and Energy Income Fund, First Trust New Opportunities MLP
& Energy Fund and First Trust Intermediate Duration Preferred & Income Fund was
held on April 22, 2016 (the "Annual Meeting"). At the Annual Meeting, James A.
Bowen and Niel B. Nielson were elected by the Common Shareholders of the First
Trust Energy Infrastructure Fund as a Class III Trustees for a three-year term
expiring at the Fund's Annual Meeting of Shareholders in 2019. The number of
votes cast for Mr. Bowen was 15,200,150, the number of votes against was 254,313
and the number of broker non-votes was 2,095,773. The number of votes cast for
Mr. Nielson was 15,190,444, the number of votes against was 264,019 and the
number of broker non-votes was 2,095,773. Richard E. Erickson, Thomas R. Kadlec,
and Robert F. Keith are the other current and continuing Trustees.

                              RISK CONSIDERATIONS

Risks are inherent in all investing. The following summarizes some, but not all,
of the risks that should be considered for the Fund. For additional information
about the risks associated with investing in the Fund, please see the Fund's
prospectus and statement of additional information, as well as other Fund
regulatory filings.

INDUSTRY CONCENTRATION RISK: The Fund invests at least 80% of its Managed Assets
in securities issued by Energy Infrastructure Companies. Given this industry
concentration, the Fund is more susceptible to adverse economic or regulatory
occurrences affecting that industry than an investment company that is not
concentrated in a single industry. Energy Infrastructure Company issuers may be
subject to a variety of factors that may adversely affect their business or
operations, including high interest costs in connection with capital
construction programs, high leverage costs associated with environmental and
other regulations, the effects of economic slowdown, surplus capacity, increased
competition from other providers of services, uncertainties concerning the
availability of fuel at reasonable prices, the effects of energy conservation
policies and other factors.

CURRENCY RISK: The value of securities denominated or quoted in foreign
currencies may be adversely affected by fluctuations in the relative currency
exchange rates and by exchange control regulations. The Fund's investment
performance may be negatively affected by a devaluation of a currency in which
the Fund's investments are denominated or quoted. Further, the Fund's investment
performance may be significantly affected, either positively or negatively, by
currency exchange rates because the U.S. dollar value of securities denominated
or quoted in another currency will increase or decrease in response to changes
in the value of such currency in relation to the U.S. dollar. While certain of
the Fund's non-U.S. dollar-denominated securities may be hedged into U.S.
dollars, hedging may not alleviate all currency risks.

INVESTMENT AND MARKET RISK: An investment in the Fund's Common Shares is subject
to investment risk, including the possible loss of the entire principal
invested. An investment in Common Shares represents an indirect investment in
the securities owned by the Fund. The value of these securities, like other
market investments, may move up or down, sometimes rapidly and unpredictably.
Common Shares at any point in time may be worth less than the original
investment, even after taking into account the reinvestment of Fund dividends
and distributions. Security prices can fluctuate for several reasons including
the general condition of the securities markets, or when political or economic
events affecting the issuers occur. When the Advisor or Sub-Advisor determines
that it is temporarily unable to follow the Fund's investment strategy or that
it is impractical to do so (such as when a market disruption event has occurred
and trading in the securities is extremely limited or absent), the Fund may take
temporary defensive positions.

LEVERAGE RISK: The use of leverage results in additional risks and can magnify
the effect of any losses. The funds borrowed pursuant to a leverage borrowing
program constitute a substantial lien and burden by reason of their prior claim
against the income of the Fund and against the net assets of the Fund in
liquidation. If the Fund is not in compliance with certain credit facility
provisions, the Fund may not be permitted to declare dividends or other
distributions.


Page 24





--------------------------------------------------------------------------------
ADDITIONAL INFORMATION (CONTINUED)
--------------------------------------------------------------------------------

                  FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)
                            MAY 31, 2016 (UNAUDITED)


MLP RISK: An investment in MLP units involves risks which differ from an
investment in common stock of a corporation. Holders of MLP units have limited
control and voting rights on matters affecting the partnership. In addition,
there are certain tax risks associated with an investment in MLP units and
conflicts of interest exist between common unit holders and the general partner,
including those arising from incentive distribution payments.

NON-U.S. RISK: The Fund may invest a portion of its assets in the equity
securities of issuers domiciled in jurisdictions other than the U.S. Investments
in the securities and instruments of non-U.S. issuers involve certain
considerations and risks not ordinarily associated with investments in
securities and instruments of U.S. issuers. Non-U.S. companies are not generally
subject to uniform accounting, auditing and financial standards and requirements
comparable to those applicable to U.S. companies. Non-U.S. securities exchanges,
brokers and listed companies may be subject to less government supervision and
regulation than exists in the United States. Dividend and interest income may be
subject to withholding and other non-U.S. taxes, which may adversely affect the
net return on such investments. A related risk is that there may be difficulty
in obtaining or enforcing a court judgment abroad.

QUALIFIED DIVIDEND INCOME TAX RISK: There can be no assurance as to what portion
of the distributions paid to the Fund's Common Shareholders will consist of
tax-advantaged qualified dividend income. Certain distributions designated by
the Fund as derived from qualified dividend income will be taxed in the hands of
non-corporate Common Shareholders at the rates applicable to long-term capital
gains, provided certain holding period and other requirements are satisfied by
both the Fund and the Common Shareholders. Additional requirements apply in
determining whether distributions by foreign issuers should be regarded as
qualified dividend income. Certain investment strategies of the Fund will limit
the Fund's ability to meet these requirements and consequently will limit the
amount of qualified dividend income received and distributed by the Fund. A
change in the favorable provisions of the federal tax laws with respect to
qualified dividends may result in a widespread reduction in announced dividends
and may adversely impact the valuation of the shares of dividend-paying
companies.

RESTRICTED SECURITIES RISK: The Fund may invest in unregistered or otherwise
restricted securities. The term "restricted securities" refers to securities
that are unregistered or are held by control persons of the issuer and
securities that are subject to contractual restrictions on their resale. As a
result, restricted securities may be more difficult to value and the Fund may
have difficulty disposing of such assets either in a timely manner or for a
reasonable price. In order to dispose of an unregistered security, the Fund,
where it has contractual rights to do so, may have to cause such security to be
registered. A considerable period may elapse between the time the decision is
made to sell the security and the time the security is registered so that the
Fund could sell it. Contractual restrictions on the resale of securities vary in
length and scope and are generally the result of a negotiation between the
issuer and acquirer of the securities. The Fund would, in either case, bear
market risks during that period.

ENERGY INFRASTRUCTURE COMPANIES RISK. The Fund invests in Energy Infrastructure
Companies and is subject to certain risks inherent in investing in these types
of securities. Energy Infrastructure Companies may be directly affected by
energy commodity prices, especially those Energy Infrastructure Companies which
own the underlying energy commodity. A decrease in the production or
availability of natural gas, natural gas liquids ("NGLs"), crude oil, coal or
other energy commodities or a decrease in the volume of such commodities
available for transportation, processing, storage or distribution may adversely
impact the financial performance of Energy Infrastructure Companies. Energy
Infrastructure Companies are subject to significant federal, state and local
government regulation in virtually every aspect of their operations, including
how facilities are constructed, maintained and operated, environmental and
safety controls, and the prices they may charge for products and services.
Various governmental authorities have the power to enforce compliance with these
regulations and the permits issued under them and violators are subject to
administrative, civil and criminal penalties, including civil fines, injunctions
or both. Stricter laws, regulations or enforcement policies could be enacted in
the future which would likely increase compliance costs and may adversely affect
the financial performance of Energy Infrastructure Companies. Natural disasters,
such as hurricanes in the Gulf of Mexico, also may impact Energy Infrastructure
Companies.

Certain Energy Infrastructure Companies are subject to the imposition of rate
caps, increased competition due to deregulation, the difficulty in obtaining an
adequate return on invested capital or in financing large construction projects,
the limitations on operations and increased costs and delays attributable to
environmental considerations, and the capital market's ability to absorb utility
debt. In addition, taxes, government regulation, international politics, price
and supply fluctuations, volatile interest rates and energy conservation may
cause difficulties for these companies. Such issuers have been experiencing
certain of these problems to varying degrees.

EQUITY SECURITIES RISK. Because the Fund invests in equity securities, the value
of the Fund's shares will fluctuate with changes in the value of these equity
securities. Equity securities prices fluctuate for several reasons, including
changes in investors' perceptions of the financial condition of an issuer or the
general condition of the relevant stock market, such as market volatility, or
when political or economic events affecting the issuers occur. In addition,
common stock prices may be particularly sensitive to rising interest rates, as
the cost of the capital rises and borrowing costs increase.


                                                                         Page 25





                      This Page Left Blank Intentionally.





                      This Page Left Blank Intentionally.





                      This Page Left Blank Intentionally.





FIRST TRUST

INVESTMENT ADVISOR
First Trust Advisors L.P.
120 E. Liberty Drive, Suite 400
Wheaton, IL  60187

INVESTMENT SUB-ADVISOR
Energy Income Partners, LLC
49 Riverside Avenue
Westport, CT 06880

ADMINISTRATOR,
FUND ACCOUNTANT &
TRANSFER AGENT
BNY Mellon Investment Servicing (US) Inc.
301 Bellevue Parkway
Wilmington, DE 19809

CUSTODIAN
The Bank of New York Mellon
101 Barclay Street, 20th Floor
New York, NY 10286

INDEPENDENT REGISTERED
PUBLIC ACCOUNTING FIRM
Deloitte & Touche LLP
111 S. Wacker Drive
Chicago, IL 60606

LEGAL COUNSEL
Chapman and Cutler LLP
111 W. Monroe Street
Chicago, IL 60603





[BLANK BACK COVER]





ITEM 2. CODE OF ETHICS.

Not applicable.

ITEM 3. AUDIT COMMITTEE FINANCIAL EXPERT.

Not applicable.

ITEM 4. PRINCIPAL ACCOUNTANT FEES AND SERVICES.

Not applicable.

ITEM 5. AUDIT COMMITTEE OF LISTED REGISTRANTS.

Not applicable.

ITEM 6. INVESTMENTS.

(a)   Schedule of Investments in securities of unaffiliated issuers as of the
      close of the reporting period is included as part of the report to
      shareholders filed under Item 1 of this form.

(b)   Not applicable.

ITEM 7. DISCLOSURE OF PROXY VOTING POLICIES AND PROCEDURES FOR CLOSED-END
MANAGEMENT INVESTMENT COMPANIES.

Not applicable.

ITEM 8. PORTFOLIO MANAGERS OF CLOSED-END MANAGEMENT INVESTMENT COMPANIES.

(a)   Not applicable.

(b)   Effective May 2, 2016, John K. Tysseland, Senior Analyst and Principal of
      Energy Income Partners, was added as a Portfolio Manager of the Fund.
      Prior to joining Energy Income Partners, Mr. Tysseland worked at Citi
      Research most currently serving as a Managing Director where he covered
      midstream energy companies and MLPs. From 1998 to 2005, he worked at
      Raymond James & Associates as a Vice President who covered the oilfield
      service industry and established the firm's initial coverage of MLPs in
      2001. Prior to that, he was an Equity Trader at Momentum Securities from
      1997 to 1998 and an Assistant Executive Director at Sumar Enterprises from
      1996 to 1997.

ITEM 9. PURCHASES OF EQUITY SECURITIES BY CLOSED-END MANAGEMENT INVESTMENT
COMPANY AND AFFILIATED PURCHASERS.

Not applicable.

ITEM 10. SUBMISSION OF MATTERS TO A VOTE OF SECURITY HOLDERS.

There have been no material changes to the procedures by which the shareholders
may recommend nominees to the Registrant's board of trustees, where those
changes were implemented after the Registrant last provided disclosure in
response to the requirements of Item 407(c)(2)(iv) of Regulation S-K (17 CFR
229.407) (as required by Item 22(b)(15) of Schedule 14A (17 CFR 240.14a-101)),
or this Item.

ITEM 11. CONTROLS AND PROCEDURES.

(a)   The Registrant's principal executive and principal financial officers, or
      persons performing similar functions, have concluded that the Registrant's
      disclosure controls and procedures (as defined in Rule 30a-3(c) under the
      Investment Company Act of 1940, as amended (the "1940 Act") (17 CFR
      270.30a-3(c))) are effective, as of a date within 90 days of the filing
      date of the report that includes the disclosure required by this
      paragraph, based on their evaluation of these controls and procedures
      required by Rule 30a-3(b) under the 1940 Act (17 CFR 270.30a-3(b)) and
      Rules 13a-15(b) or 15d-15(b) under the Securities Exchange Act of 1934, as
      amended (17 CFR 240.13a-15(b) or 240.15d-15(b)).

(b)   There were no changes in the Registrant's internal control over financial
      reporting (as defined in Rule 30a-3(d) under the 1940 Act (17 CFR
      270.30a-3(d)) that occurred during the Registrant's second fiscal quarter
      of the period covered by this report that has materially affected, or is
      reasonably likely to materially affect, the Registrant's internal control
      over financial reporting.

ITEM 12. EXHIBITS.

(a)(1) Not applicable.

(a)(2) Certifications pursuant to Rule 30a-2(a) under the 1940 Act and Section
       302 of the Sarbanes-Oxley Act of 2002 are attached hereto.

(a)(3) Not applicable.

(b)    Certifications pursuant to Rule 30a-2(b) under the 1940 Act and Section
       906 of the Sarbanes- Oxley Act of 2002 are attached hereto.

(c)    Notices to the registrant's common shareholders in accordance with the
       order under Section 6(c) of the 1940 Act granting an exemption from
       Section 19(b) of the 1940 Act and Rule 19a-1 under the 1940 Act, dated
       March 24, 2010. (1)

      (1)   The Fund received exemptive relief from the Securities and Exchange
            Commission which permits the Fund to make periodic distributions of
            long-term capital gains as frequently as monthly each taxable year.
            The relief is conditioned, in part, on an undertaking by the Fund to
            make the disclosures to the holders of the Fund's common shares, in
            addition to the information required by Section 19(a) of the 1940
            Act and Rule 19a-1 thereunder. The Fund is likewise obligated to
            file with the SEC the information contained in any such notice to
            shareholders. In that regard, attached as an exhibit to this filing
            is a copy of such notice made during the period.





                                   SIGNATURES

Pursuant to the requirements of the Securities Exchange Act of 1934 and the
Investment Company Act of 1940, the registrant has duly caused this report to be
signed on its behalf by the undersigned, thereunto duly authorized.

(registrant)        First Trust Energy Infrastructure Fund
              ---------------------------------------------------

By (Signature and Title)*               /s/ James M. Dykas
                                        ----------------------------------------
                                        James M. Dykas, President and
                                        Chief Executive Officer
                                        (principal executive officer)

Date: July 19, 2016
     ---------------

Pursuant to the requirements of the Securities Exchange Act of 1934 and the
Investment Company Act of 1940, this report has been signed below by the
following persons on behalf of the registrant and in the capacities and on the
dates indicated.

By (Signature and Title)*               /s/ James M. Dykas
                                        ----------------------------------------
                                        James M. Dykas, President and
                                        Chief Executive Officer
                                        (principal executive officer)

Date: July 19, 2016
     ---------------

By (Signature and Title)*               /s/ Donald P. Swade
                                        ----------------------------------------
                                        Donald P. Swade, Treasurer,
                                        Chief Financial Officer and
                                        Chief Accounting Officer
                                        (principal financial officer)

Date: July 19, 2016
     ---------------

* Print the name and title of each signing officer under his or her signature.