x^3 e ^ x^2 d x - Vidyadip

Question

$\int x^3 e ^{x^2} d x$

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Answer

$ \text { Let } I =\int x^3 \cdot e ^{x^2} d x$
$=\int x^2 \cdot x e ^{x^2} d x $
Put $x^2=t$
$ \therefore 2 x \cdot dx = dt$
$\therefore x dx =\frac{ dt }{2}$
$\therefore I =\frac{1}{2} \int te ^{ t } dt$
$=\frac{1}{2}\left[ t \int e ^{ t } dt -\int\left[\frac{ d }{ dt }( t ) \int e ^{ t } dt \right] dt \right]$
$=\frac{1}{2}\left[ te ^{ t }-\int 1 \cdot e ^{ t } dt \right]$
$=\frac{1}{2}\left( te ^{ t }- e ^{ t }\right)+ c $
$=\frac{1}{2} e ^{ t }( t -1)+ c$
$\therefore I =\frac{1}{2} e ^{x^2}\left(x^2-1\right)+ c $

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