Mathematics

# $\int {x^2}\,{e^x}\, dx =?$

##### SOLUTION
$I=\displaystyle\int{{x}^{2}{e}^{x}dx}$

Integrating by parts, we get

Let $u={x}^{2}\Rightarrow\,du=2x\,dx$

$dv={e}^{x}dx\Rightarrow\,v={e}^{x}$

$I={x}^{2}{e}^{x}-\displaystyle\int{2x{e}^{x}dx}$

$I={x}^{2}{e}^{x}-2\displaystyle\int{x{e}^{x}dx}$

Let $u=x\Rightarrow\,du=dx$

$dv={e}^{x}dx\Rightarrow\,v={e}^{x}$

$I={x}^{2}{e}^{x}-2\left[x{e}^{x}-\displaystyle\int{{e}^{x}dx}\right]$

$I={x}^{2}{e}^{x}-2\left[x{e}^{x}-{e}^{x}\right]+c$

$\therefore\,I={x}^{2}{e}^{x}-2x{e}^{x}+2{e}^{x}+c$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

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