Notice Regarding Your Monthly Distribution

                  FIRST TRUST ENERGY INFRASTRUCTURE FUND (FIF)

The closed-end fund listed above has declared a distribution payable on June 15,
2016, to shareholders of record as of June 3, 2016, with an ex-dividend date of
June 1, 2016. This Notice is meant to provide you information about the sources
of your Fund's distributions. You should not draw any conclusions about the
Fund's investment performance from the amount of its distribution or from the
terms of its Managed Distribution Plan.

The following tables set forth the estimated amounts of the current distribution
and the cumulative distributions paid this fiscal year to date for the Fund from
the following sources: net investment income ("NII"); net realized short-term
capital gains ("STCG"); net realized long-term capital gains ("LTCG"); and
return of capital ("ROC"). These estimates are based upon information as of May
31, 2016, are calculated based on a generally accepted accounting principles
("GAAP") basis and include the prior fiscal year-end undistributed net
investment income. The amounts and sources of distributions are expressed per
common share.



                                                                                                              ANNUALIZED    5 YEAR
                                                                                                               CURRENT       AVG.
                               TOTAL                                                                            DIST.       ANNUAL
                              CURRENT         CURRENT DISTRIBUTION ($)           CURRENT DISTRIBUTION (%)     DIST. RATE    TOTAL
 FUND      FUND     FISCAL    DISTRIBU- ------------------------------------  ------------------------------  AS A % OF     RETURN
TICKER    CUSIP    YEAR END     TION      NII       STCG      LTCG    ROC (2)   NII    STCG    LTCG   ROC (2)   NAV (3)   ON NAV (4)
------- --------- ---------- ---------- --------  --------  --------  ------  ------  ------  ------  ------  ----------  ----------
                                                                                      
FIF (5) 33738C103 11/30/2016 $  0.11000 $0.02312      --       --    $0.08688 21.01%    --      --    78.99%     7.04%      10.76%

                                                                                                              CUMULATIVE
                                TOTAL                                                                         FISCAL YTD  CUMULATIVE
                             CUMULATIVE       CUMULATIVE DISTRIBUTIONS           CUMULATIVE DISTRIBUTIONS     DISTRIBU-   FISCAL YTD
                             FISCAL YTD            FISCAL YTD ($)                     FISCAL YTD (%)            TIONS       TOTAL
 FUND      FUND     FISCAL    DISTRIBU- ------------------------------------  ------------------------------  AS A % OF     RETURN
TICKER    CUSIP    YEAR END   TIONS (1)    NII      STCG       LTCG   ROC (2)   NII    STCG    LTCG   ROC (2)   NAV (3)   ON NAV (4)
------- --------- ---------- ---------- --------  --------  --------  ------  ------  ------  ------  ------  ----------  ----------
FIF (5) 33738C103 11/30/2016 $  1.11000 $0.09246     --     $0.67000 $0.34754  8.33%    --    60.36%  31.31%     5.92%      10.58%


(1)   Includes the most recent monthly distribution paid on June 15, 2016.

(2)   The Fund estimates that it has distributed more than its income and net
      realized capital gains; therefore, a portion of your distribution may be a
      return of capital. A return of capital may occur, for example, when some
      or all of the money that you invested in the Fund is paid back to you. A
      return of capital distribution does not necessarily reflect the Fund's
      investment performance and should not be confused with "yield" or
      "income."

(3)   Based on Net Asset Value ("NAV") as of May 31, 2016.

(4)   Total Returns are through May 31, 2016 and since the Fund's inception was
      less than five years ago, the return included in the 5 Year Average Annual
      Total Return on NAV column is from September 27, 2011, the Fund's
      inception, through May 31, 2016.

(5)   The Fund anticipates that, due to the tax treatment of cash distributions
      made by Master Limited Partnerships in which the Fund invests, a portion
      of distributions the Fund makes to Common Shareholders may consist of a
      tax-deferred return of capital.

The amounts and sources of distributions reported in this Notice are only
estimates and are not being provided for tax reporting purposes. The actual
amounts and sources of the amounts for tax reporting purposes will depend upon
the Fund's investment experience during the remainder of its fiscal year and may
be subject to changes based on tax regulations. The Fund will send you a Form
1099-DIV for the calendar year that will tell you how to report these
distributions for federal income tax purposes. You should not use this Notice as
a substitute for your Form 1099-DIV.

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First Trust Advisors L.P. Contact:
     Jim Dykas (630) 517-7665