ToWe evaluate aim the integral to:

evaluate:

[ \I[ =\int \int 99xx^^2 e2 e^{-^{-22xx}} , , dx dx , ]

]

Stepwe will 1 use: ** Useintegration Integration by parts by Parts** Twice repeatedly.

Integration---

by Step parts formula 1:

:[ General Formula\ forint Integration u by \ Parts, dv =The formula uv for - integration \int by parts v \ is: , du. [ \]

int u#### \ First, Application dv =Let ( uv - u \ = int v9 \x,^2 du . ) and]

(We will dv de =compose e^{- (2 9xx}^ \2, e dx^{- \2).x Then}: \)[ intou parts = such that9 repeatedx applications^ of2 this, \ formula simplifyquad du the integral =.

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18###x Step , 2 dx:, First \ Applicationquad of v Integration = by \ Partsint eLet^{-: 2\x[ } \u, = dx9 =x -^\2frac {quad1}{ \2text}{ eand}^{-2 \quadx}. dv\ =]

eSub^{-stitute2x into} the formula : ,\ dx[ . \]

intThen 9: x^2du e = ^{-218xx \} \,, dx \ dxquad = - \text\{fracand{9}x \^quad2 v =}{2 \int} e e^{-^{-22xx}} + \, \ dxint = \ \fracfrac{{-18x1}{}{22}} e e^{-^{-22xx}}. \

,Sub dxstitute. into\ the] integrationSimpl byify parts: formula: [ $$intI =9x uv -^ \2int e v^{- \2x, du}, \,] dx\ = -[ I\ =frac {left9x(^ 29}{x^22} \ e^{-cdot2 \xfrac}{- + 1}{92 }int e x^{- e^{-2x2}x} \right ), - dx . int$$

left####( Second \ Applicationfrac {-For1 }{ \2int} x e e^{-^{-22xx}} \ \cd, dxot \ 18), letx \ \(, u dx = \ xright. )\ and (]

Simpl dvify =: e^{-[ 2Ix =} \ -,\frac dx{ \9).x Then^: 2[ }{u2 =} x e,^{-2 \xquad} du + = dx ,9 \ \quadint v x e = -^{-\2fracx{} \1}{, dx2}. e^{-]

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2###x Step}. \3]

:Sub Secondstitute into Application of the Integration formula by: \ Parts[

\Weint now x focus on e (^{-2 \xint x} e ^{-, dx2x = -}\ ,frac dx{x }{).2 Let}: e^{-2[ ux =} + x \ \quadfrac{ \text1{}{2and}} \ \intquad e dv =^{-2 e^{-x2}x ,} dx . ,\ dx] . \The last]

integralThen is: : $$[ du\ =int dx e ^{-quad \2textx{}and }, \ dx =quad - v =\ \fracfrac{1{-1}{2}{2} e} e^{-^{-2x2}. x}. $$ Thus]

Sub: \stitute[ into\ the integrationint by x parts e formula^{-2: x}[ \ ,int x dx e = -^{-2\fracx{x}}{ ,2 dx} e = ^{-left2(x x} - \ \cdotfrac{ \1frac{-}{14}{}2 e}^{- e2^{-x2}. x}]

####right Combine) Results - \Subintstitute \ backleft: (\ frac\{-int 1}{92x}^ e2^{- e^{-22x}x \}, \ dx, \ dxright =). -\

fracSimpl{9ify: x^[ 2}{int2 x} e e^{-^{-22xx}} + , dx9 = \ -left(frac -{xfrac}{{x2}{} e2}^{- e2x^{-2}x + }frac -{ \1frac{}{41}}{ e4}^{-2 e^{-x2}. \x]

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\right### Step). \4]

:Simpl Substituteify Back:

\Now[ substitute\ (int \ int9 xx e^2^{- e2^{-x2} \x,} dx \ ,) dx = into the - previous\frac result: