Integrate the function x^2e^x - Vidyadip

Question

Integrate the function $x^2e^x$

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Answer

$\int {{x^2}{e^x}} dx$$ = {x^2}\int {{e^x}dx - \int {\left[ {\frac{d}{{dx}}{x^2}\int {{e^x}dx} } \right]} dx} $
[Applying product rule]
$ = {x^2}{e^x} - \int {2x{e^x}dx} $
$ = {x^2}{e^x} - 2\int {x{e^x}dx} $
$= {x^2}{e^x} - 2\left[ {{xe^x}dx - \int {\left\{ {\frac{d}{{dx}}x\int {{e^x}dx} } \right\}dx} } \right]$
[Again applying product rule]
$= {x^2}{e^x} - 2\left( {x{e^x} - \int {1.{e^x}dx} } \right)$
$= {x^2}{e^x} - 2\left( {x{e^x} - \int {{e^x}dx} } \right)$
$ = {x^2}{e^x} - 2x{e^x} + 2\int {{e^x}dx} $
$= x^2e^x - 2xe^x + 2e^x + c$
$= e^x(x^2 - 2x + 2) + c$

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