Question

Integrate the function $x^3{e}^x$

Answer 1

Let $I=\int x^2{e}^{x}dx$
Taking $x^2$ as first function and $e^x$ as second function and integrating using by parts, we obtain
$I=x^2\int e^{x}dx-\int \{\left(\frac{d}{dx}{x}^2\right)\int e^{x}dx\}dx$
$=x^2{e}^x-\int 2x\cdot e^{x}dx$
$=x^2{e}^x-2[x\cdot \int e^{x}dx-\int \{\left(\frac{d}{dx}x\right)\cdot \int e^{x}dx\}dx]$
$=x^2{e}^x-2[x{e}^x-\int e^{x}dx]$
$=x^2{e}^x-2[x{e}^x-e^x]$
$=x^2{e}^x-2x{e}^x+2e^x+C$
$=e^x(x^2-2x+2)+C$