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

                                    FORM N-Q

       QUARTERLY SCHEDULE OF PORTFOLIO HOLDINGS OF REGISTERED MANAGEMENT
                               INVESTMENT COMPANY

                  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: February 29, 2012
                                           -------------------

Form N-Q is to be used by management investment companies, other than small
business investment companies registered on Form N-5 (ss.ss. 239.24 and 274.5 of
this chapter), to file reports with the Commission, not later than 60 days after
the close of the first and third fiscal quarters, pursuant to rule 30b1-5 under
the Investment Company Act of 1940 (17 CFR 270.30b1-5). The Commission may use
the information provided on Form N-Q in its regulatory, disclosure review,
inspection, and policymaking roles.

A registrant is required to disclose the information specified by Form N-Q, and
the Commission will make this information public. A registrant is not required
to respond to the collection of information contained in Form N-Q 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 the
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. SCHEDULE OF INVESTMENTS. The Schedule(s) of Investments is attached
herewith.


FIRST TRUST ENERGY INFRASTRUCTURE FUND
PORTFOLIO OF INVESTMENTS (A)
FEBRUARY 29, 2012 (UNAUDITED)



    SHARES/
     UNITS                                      DESCRIPTION                                    VALUE
---------------  --------------------------------------------------------------------      --------------
MASTER LIMITED PARTNERSHIPS - 31.8%

                 GAS UTILITIES - 1.2%
                                                                                     
        107,664  AmeriGas Partners, L.P. (b).........................................      $    4,868,566
                                                                                           --------------

                 OIL, GAS & CONSUMABLE FUELS - 30.6%
         20,000  Alliance GP Holdings, L.P. .........................................           1,006,200
         54,700  Alliance Resource Partners, L.P. ...................................           3,925,819
        130,460  Buckeye Partners, L.P. (b)..........................................           7,801,508
        112,000  El Paso Pipeline Partners, L.P. ....................................           4,107,040
        124,000  Energy Transfer Equity, L.P. (b)....................................           5,392,760
         31,600  Enterprise Products Partners, L.P. (b)..............................           1,639,408
        116,223  Holly Energy Partners, L.P. (b).....................................           7,116,334
         89,300  Magellan Midstream Partners, L.P. (b)...............................           6,534,081
        348,952  Natural Resource Partners, L.P. (b).................................           8,517,918
        199,000  NuStar Energy, L.P. (b).............................................          12,099,200
         29,600  NuStar GP Holdings, LLC ............................................           1,029,784
         16,000  ONEOK Partners, L.P. ...............................................             931,200
        144,443  Plains All American Pipeline, L.P. (b)..............................          11,945,436
        102,700  Spectra Energy Partners, L.P. ......................................           3,388,073
        245,409  Sunoco Logistics Partners, L.P. (b).................................           9,583,222
        353,095  TC Pipelines, L.P. (b)..............................................          16,397,732
        289,828  Teekay LNG Partners, L.P. (b).......................................          11,361,258
        231,570  TransMontaigne Partners, L.P. (b)...................................           8,042,426
         41,300  Williams Partners, L.P. ............................................           2,569,273
                                                                                           --------------
                                                                                              123,388,672
                                                                                           --------------
                 TOTAL MASTER LIMITED PARTNERSHIPS ..................................         128,257,238
                 (Cost $112,303,926)                                                       --------------

COMMON STOCKS - 97.1%

                 ELECTRIC UTILITIES - 21.3%
        385,100  Duke Energy Corp. (b)...............................................           8,056,292
        274,600  Emera, Inc. (CAD) (b)...............................................           9,250,006
        296,450  Exelon Corp. (b)....................................................          11,582,301
         89,800  Fortis, Inc. (CAD) (b)..............................................           2,962,325
         70,000  ITC Holdings Corp. (b)..............................................           5,283,600
        301,200  NextEra Energy, Inc. (b)............................................          17,924,412
        438,700  Northeast Utilities (b).............................................          15,749,330
        340,000  Southern Co. (b)....................................................          15,024,600
                                                                                           --------------
                                                                                               85,832,866
                                                                                           --------------

                 GAS UTILITIES - 5.0%
         15,800  ONEOK, Inc. (b).....................................................           1,305,712
        260,000  Questar Corp. (b)...................................................           4,997,200
        492,600  UGI Corp. (b).......................................................          13,915,950
                                                                                           --------------
                                                                                               20,218,862
                                                                                           --------------

                 MULTI-UTILITIES - 17.6%
         35,000  Atco, Ltd. (CAD) (b)................................................           2,399,717
         45,500  Canadian Utilities, Ltd. (CAD) (b)..................................           3,032,260
        807,600  Centerpoint Energy, Inc. (b)........................................          15,740,124
        366,400  Dominion Resources, Inc. (b)........................................          18,492,208


                See Notes to Quarterly Portfolio of Investments           Page 1



FIRST TRUST ENERGY INFRASTRUCTURE FUND
PORTFOLIO OF INVESTMENTS (a) - (CONTINUED)
FEBRUARY 29, 2012 (UNAUDITED)


     SHARES                                     DESCRIPTION                                    VALUE
---------------  --------------------------------------------------------------------      --------------

COMMON STOCKS - (CONTINUED)

                 MULTI-UTILITIES - (CONTINUED)
        175,000  National Grid PLC, ADR (b)..........................................      $    8,958,250
        708,300  NiSource, Inc. (b)..................................................          16,999,200
         50,000  Sempra Energy (b)...................................................           2,962,000
         75,000  Wisconsin Energy Corp. .............................................           2,556,000
                                                                                           --------------
                                                                                               71,139,759
                                                                                           --------------

                 OIL, GAS & CONSUMABLE FUELS - 53.2%
        639,700  El Paso Corp. (b)...................................................          17,790,057
        189,969  Enbridge Energy Management, LLC (b) (c).............................           6,415,253
        860,684  Enbridge Energy Management, LLC (c) (d) (e).........................          28,832,914
        549,600  Enbridge Income Fund Holdings, Inc. (CAD) (b).......................          12,542,289
        380,169  Enbridge, Inc. (b)..................................................          14,678,325
        252,300  Keyera Corp. (CAD) (b)..............................................          10,760,584
        493,281  Kinder Morgan Management, LLC (b) (c)...............................          39,546,323
        292,300  Kinder Morgan, Inc. (b).............................................          10,300,652
        482,100  Pembina Pipeline Corp. (CAD) (b)....................................          13,676,838
        428,850  Spectra Energy Corp. (b)............................................          13,457,313
        480,400  TransCanada Corp. (b)...............................................          21,127,992
        258,000  Veresen, Inc. (CAD) (b).............................................           3,976,452
        730,700  Williams Cos., Inc. (b).............................................          21,833,316
                                                                                           --------------
                                                                                              214,938,308
                                                                                           --------------
                 TOTAL COMMON STOCKS ................................................         392,129,795
                 (Cost $345,918,185)                                                       --------------

                 TOTAL INVESTMENTS - 128.9% .........................................         520,387,033
                 (Cost $458,222,111) (f)                                                   --------------

   NUMBER OF
   CONTRACTS                                    DESCRIPTION                                    VALUE
---------------  --------------------------------------------------------------------      --------------

CALL OPTIONS WRITTEN - (0.5%)
                 Centerpoint Energy, Inc., Call
          2,000  @ 20 due May 12 ....................................................            (120,000)
                                                                                           --------------
                 Dominion Resources, Inc., Calls
            975  @ 55 due April 12 ..................................................              (4,875)
          1,853  @ 55 due July 12 ...................................................             (48,178)
                                                                                           --------------
                                                                                                  (53,053)
                                                                                           --------------
                 Duke Energy Corp., Calls
             91  @ 21 due April 12 ..................................................              (3,185)
            316  @ 22 due April 12 ..................................................              (3,160)
            400  @ 23 due July 12 ...................................................              (3,000)
                                                                                           --------------
                                                                                                   (9,345)
                                                                                           --------------
                 El Paso Corp., Calls
          2,000  @ 26 due April 12 ..................................................            (420,000)
          1,800  @ 27 due April 12 ..................................................            (225,900)
             97  @ 28 due April 12 ..................................................              (4,850)
                                                                                           --------------
                                                                                                 (650,750)
                                                                                           --------------

Page 2          See Notes to Quarterly Portfolio of Investments


FIRST TRUST ENERGY INFRASTRUCTURE FUND
PORTFOLIO OF INVESTMENTS (a) - (CONTINUED)
FEBRUARY 29, 2012 (UNAUDITED)


   NUMBER OF
   CONTRACTS                                    DESCRIPTION                                    VALUE
---------------  --------------------------------------------------------------------      --------------

CALL OPTIONS WRITTEN - (CONTINUED)
                 Enbridge, Inc., Call
          2,280  @ 40 due April 12 ..................................................      $     (102,600)
                                                                                           --------------
                 Exelon Corp., Calls
          1,379  @ 41 due March 12 ..................................................              (6,895)
            183  @ 42 due July 12 ...................................................              (4,941)
                                                                                           --------------
                                                                                                  (11,836)
                                                                                           --------------
                 Kinder Morgan, Inc., Calls
            800  @ 35 due March 12 ..................................................             (60,000)
            700  @ 35 due September 12 ..............................................            (122,500)
                                                                                           --------------
                                                                                                 (182,500)
                                                                                           --------------
                 National Grid PLC, Call
            550  @ 50 due March 12 ..................................................             (83,875)
                                                                                           --------------
                 NextEra Energy, Inc., Calls
            200  @ 60 due March 12 ..................................................              (9,000)
          1,086  @ 65 due June 12 ...................................................             (10,860)
                                                                                           --------------
                                                                                                  (19,860)
                                                                                           --------------
                 NiSource, Inc., Calls
          3,200  @ 25 due April 12 ..................................................            (176,000)
            883  @ 25 due July 12 ...................................................             (79,470)
                                                                                           --------------
                                                                                                 (255,470)
                                                                                           --------------
                 Northeast Utilities, Calls
            129  @ 35 due March 12 ..................................................             (14,190)
            700  @ 35 due April 12 ..................................................             (80,500)
                                                                                           --------------
                                                                                                  (94,690)
                                                                                           --------------
                 Questar Corp., Calls
            850  @ 20 due March 12 ..................................................              (8,500)
            714  @ 21 due July 12 ...................................................             (10,710)
                                                                                           --------------
                                                                                                  (19,210)
                                                                                           --------------
                 Southern Co., Calls
          1,379  @ 46 due May 12 ....................................................             (30,338)
             95  @ 47 due August 12 .................................................              (2,945)
                                                                                           --------------
                                                                                                  (33,283)
                                                                                           --------------
                 Spectra Energy Corp., Calls
            500  @ 30 due March 12 ..................................................             (78,500)
            800  @ 31 due March 12 ..................................................             (56,000)
             73  @ 32 due March 12 ..................................................              (1,825)
          1,200  @ 33 due June 12 ...................................................             (48,000)
                                                                                           --------------
                                                                                                 (184,325)
                                                                                           --------------
                 TransCanada Corp., Calls
          2,186  @ 45 due May 12 ....................................................            (174,880)
            500  @ 45 due August 12 .................................................             (62,500)
                                                                                           --------------
                                                                                                 (237,380)
                                                                                           --------------

                See Notes to Quarterly Portfolio of Investments           Page 3



FIRST TRUST ENERGY INFRASTRUCTURE FUND
PORTFOLIO OF INVESTMENTS (a) - (CONTINUED)
FEBRUARY 29, 2012 (UNAUDITED)


    NUMBER OF
    CONTRACTS                                   DESCRIPTION                                    VALUE
---------------  --------------------------------------------------------------------      --------------

CALL OPTIONS WRITTEN - (CONTINUED)
                 UGI Corp., Call
          1,000  @ 30 due March 12 ..................................................      $      (10,000)
                                                                                           --------------
                 Williams Cos., Inc., Calls
          1,000  @ 30 due March 12 ..................................................             (40,000)
          2,306  @ 31 due March 12 ..................................................             (23,060)
          1,000  @ 32 due March 12 ..................................................              (3,500)
            320  @ 35 due May 12 ....................................................              (2,240)
                                                                                           --------------
                                                                                                  (68,800)
                                                                                           --------------
                 TOTAL CALL OPTIONS WRITTEN .........................................          (2,136,977)
                 (Premiums received $1,353,809)                                            --------------

                 OUTSTANDING LOAN - (30.5%) .........................................        (123,000,000)

                 NET OTHER ASSETS AND LIABILITIES - 2.1% ............................           8,348,890
                                                                                           --------------
                 NET ASSETS - 100.0% ................................................      $  403,598,946
                                                                                           ==============


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

(a)   All percentages shown in the Portfolio of Investments are based on net
      assets.

(b)   All or a portion of this security is available to serve as collateral on
      the outstanding loan.

(c)   Non-income producing security which pays in-kind distributions.

(d)   This security is fair valued in accordance with procedures adopted by the
      Fund's Board of Trustees.

(e)   This security, sold within the terms of a private placement memorandum, is
      exempt from registration upon resale under Rule 144A under the Securities
      Act of 1933, as amended (the "1933 Act"), and may be resold in
      transactions exempt from registration, normally to qualified institutional
      buyers (see Note 1D - Restricted Securities in the Notes to Quarterly
      Portfolio of Investments).

(f)   Aggregate cost for financial reporting purposes, which approximates the
      aggregate cost for federal income tax purposes. As of February 29, 2012,
      the aggregate gross unrealized appreciation for all securities in which
      there was an excess of value over tax cost was $64,560,577 and the
      aggregate gross unrealized depreciation for all securities in which there
      was an excess of tax cost over value was $2,395,655.

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.


Page 4          See Notes to Quarterly Portfolio of Investments



FIRST TRUST ENERGY INFRASTRUCTURE FUND
PORTFOLIO OF INVESTMENTS (a) - (CONTINUED)
FEBRUARY 29, 2012 (UNAUDITED)

---------------------------------------
VALUATION INPUTS

A summary of the inputs used to value the Fund's investments as of February 29,
2012 is as follows (see Note 1A - Portfolio Valuation in the Notes to Quarterly
Portfolio of Investments):


                                                 ASSETS TABLE
                                                                                     LEVEL 2         LEVEL 3
                                                     TOTAL          LEVEL 1        SIGNIFICANT     SIGNIFICANT
                                                    VALUE AT         QUOTED        OBSERVABLE      UNOBSERVABLE
                                                   2/29/2012         PRICES          INPUTS          INPUTS
                                                 --------------  --------------  --------------  --------------

                                                                                     
Master Limited Partnerships*..................   $  128,257,238  $  128,257,238  $           --  $           --
Common Stocks:
    Oil, Gas & Consumable Fuels...............      214,938,308     186,105,394      28,832,914              --
    Other Industry Categories*................      177,191,487     177,191,487              --              --
                                                 --------------  --------------  --------------  --------------
TOTAL INVESTMENTS.............................   $  520,387,033  $  491,554,119  $   28,832,914  $           --
                                                 ==============  ==============  ==============  ==============

                                               LIABILITIES TABLE
                                                                                    LEVEL 2         LEVEL 3
                                                     TOTAL          LEVEL 1       SIGNIFICANT     SIGNIFICANT
                                                   VALUE AT         QUOTED        OBSERVABLE     UNOBSERVABLE
                                                   2/29/2012        PRICES          INPUTS          INPUTS
                                                 --------------  --------------  --------------  --------------
Call Options Written..........................   $   (2,136,977) $   (2,136,977) $           --  $           --
                                                 ==============  ==============  ==============  ==============



* See the Portfolio of Investments for the industry breakout. Industry
  categories are only shown separately if they include holdings in two or more
  levels.



                See Notes to Quarterly Portfolio of Investments           Page 5



NOTES TO QUARTERLY PORTFOLIO OF INVESTMENTS

                     FIRST TRUST ENERGY INFRASTRUCTURE FUND
                         FEBRUARY 29, 2012 (UNAUDITED)



                     1. VALUATION AND INVESTMENT PRACTICES

A. PORTFOLIO VALUATION:

The net asset value ("NAV") of the Common Shares of First Trust Energy
Infrastructure Fund (the "Fund") is determined daily as of the close of regular
trading on the New York Stock Exchange ("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. Domestic debt securities
and foreign securities are priced using data reflecting the earlier closing of
the principal markets for those securities. The 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 in accordance with valuation procedures
adopted by the Fund's Board of Trustees, and in accordance with provisions of
the Investment Company Act of 1940, as amended (the "1940 Act"). The following
securities, for which accurate and reliable market quotations are readily
available, will be valued as follows:

      Common stocks, master limited partnerships ("MLPs") and other securities
      listed on any national or foreign exchange (excluding the NASDAQ National
      Market ("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. If there are no transactions on the valuation
      day, the securities are valued at the mean between the most recent bid and
      asked prices.

      Securities listed on the NASDAQ or the AIM are valued at the official
      closing price. If there is no official closing price on the valuation day,
      the securities are valued at the mean between the most recent bid and
      asked prices.

      Securities traded in the over-the-counter market are valued at their
      closing bid prices.

      Exchange-traded options and futures 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 and futures contracts
      are valued at the mean between the most recent bid and asked prices.
      Over-the-counter options and futures contracts are valued at their closing
      bid prices.

Short-term investments that mature in less than 60 days when purchased are
valued at amortized cost.

All market quotations used in valuing the Fund's securities will be obtained
from a third party pricing service. If no quotation is received from a pricing
service, attempts will be made to obtain one or more broker quotes for the
security. In the event the pricing service does not provide a valuation, broker
quotations are not readily available, or the valuations received are deemed
unreliable, the Fund's Board of Trustees has designated First Trust Advisors
L.P. ("First Trust") to use a fair value method to value the Fund's securities.
Additionally, if events occur after the close of the principal markets for
certain securities (e.g., domestic debt and foreign securities) that could
materially affect the Fund's NAV, First Trust will use a fair value method to
value the Fund's securities. The use of fair value pricing is governed by
valuation procedures adopted by the Fund's Board of Trustees, and in accordance
with the provisions of the 1940 Act. As a general principle, the fair value of a
security is the amount which the Fund might reasonably expect to receive for the
security upon its current sale. However, in light of the judgment involved in
fair valuations, there can be no assurance that a fair value assigned to a
particular security will be the amount which the Fund might be able to receive
upon its current sale. Fair valuation of a security will be based on the
consideration of all available information, 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.


Page 6



NOTES TO FINANCIAL STATEMENTS - (CONTINUED)

                     FIRST TRUST ENERGY INFRASTRUCTURE FUND
                         FEBRUARY 29, 2012 (UNAUDITED)

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
      securities. An active market is a market in which transactions for the
      security 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 securities in active markets.

      o     Quoted prices for identical or similar securities in markets that
            are non-active. A non-active market is a market where there are few
            transactions for the security, 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 security
            (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 security.

The inputs or methodology used for valuing securities are not necessarily an
indication of the risk associated with investing in those securities. A summary
of the inputs used to value the Fund's investments as of February 29, 2012, is
included with the Fund's Portfolio of Investments.

B. OPTION CONTRACTS:

The Fund is subject to equity price risk in the normal course of pursuing its
investment objective and may enter into options written to hedge against changes
in the value of equities. Also, by writing (selling) options, the Fund seeks to
generate additional income, in the form of premiums received for writing
(selling) the options. The Fund may write (sell) covered call or put options
("options") on all or a portion of the common stock of equities 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 of equities 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.

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.

                                                                          Page 7



NOTES TO FINANCIAL STATEMENTS - (CONTINUED)

                     FIRST TRUST ENERGY INFRASTRUCTURE FUND
                         FEBRUARY 29, 2012 (UNAUDITED)

Over-the-counter ("OTC") 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. SECURITIES TRANSACTIONS:

Securities transactions are recorded as of the trade date. Realized gains and
losses from securities transactions are recorded on the identified cost basis.

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.

D. FOREIGN CURRENCY:

The books and records of the Fund are maintained in U.S. dollars. Investments
denominated in foreign currency are translated into U.S. dollars at the exchange
rates prevailing at the end of the period. Purchases and sales of investment
securities are translated on the respective dates of such transactions.

E. RESTRICTED SECURITIES:

The Fund may invest up to 15% of its Managed Assets in restricted securities.
Managed Assets means the average daily total asset value of the Fund minus the
sum of the Fund's liabilities other than the principal amount of borrowings.
Restricted securities are securities that may not be offered for public sale
without first being registered under the Securities Act of 1933, as amended (the
"1933 Act"). Prior to registration, restricted securities may only be resold in
transactions exempt from registration under Rule 144A under the 1933 Act,
normally to qualified institutional buyers. As of February 29, 2012, the Fund
held restricted securities as shown in the following table. The Fund does not
have the right to demand that such securities be registered. These securities
are valued according to the valuation procedures as stated in the Portfolio
Valuation note (Note 1A).


                                                                                                                  % OF
                                        ACQUISITION    SHARES/                   CARRYING                         NET
SECURITY                                   DATE         UNITS        PRICE         COST            VALUE         ASSETS
--------------------------------------- ----------- -------------- ---------- --------------- --------------- ------------
                                                                                                
Enbridge Energy Management, LLC         11/14/2011     860,684       $33.50    $ 25,700,000    $ 28,832,914       7.1%




                               2. OPTION ACTIVITY

Written option activity for the Fund for the period ended February 29, 2012 was
as follows:

                                                   Number
                                                     of
Written Options                                   Contracts     Premiums
---------------------------------------------    ----------    -----------
Options outstanding at November 30, 2011.....      29,526      $ 1,735,315
Options Written..............................      35,970        1,165,294
Options Expired..............................     (21,403)      (1,012,974)
Options Exercised............................      (8,034)        (426,402)
Options Closed...............................        (514)        (107,424)
                                                 ----------    -----------
Options outstanding at February 29, 2012.....      35,545      $ 1,353,809
                                                 ==========    ===========


Page 8

ITEM 2. 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 last fiscal quarter
       that have materially affected, or are reasonably likely to materially
       affect, the registrant's internal control over financial reporting.

ITEM 3. EXHIBITS.

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


                                   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/ Mark R. Bradley
                          ------------------------------------------------------
                          Mark R. Bradley, President and Chief Executive Officer
                          (principal executive officer)

Date  April 16, 2012
     ---------------------

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/ Mark R. Bradley
                          ------------------------------------------------------
                          Mark R. Bradley, President and Chief Executive Officer
                          (principal executive officer)

Date  April 16, 2012
     ---------------------

By (Signature and Title)*   /s/ James M. Dykas
                          ------------------------------------------------------
                          James M. Dykas, Treasurer, Chief Financial Officer and
                          Chief Accounting Officer
                          (principal financial officer)

Date  April 16, 2012
     ---------------------

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