Question

Integrate the function.
$\int x^2{e}^{x}dx.$

Answer 1

Let $I=\int x^2{e}^{x}dx$
On taking $x^2$ as first function and $e^x$ as second function and integrating by parts, we get $I=x^2\int e^{x}dx-\int \left[\frac{d}{dx}\right(x^2)\int e^{x}dx]dx=x^2{e}^x-\int \left[2x{e}^x\right]dx$
Again, integrating by parts, we get
$I=x^2{e}^x-\{2x\int e^{x}dx-2\int \left[\frac{d}{dx}\right(x)\int e^{x}dx]dx\}=x^2{e}^x-2x{e}^x+2\int e^{x}dx=x^2{e}^x-2x{e}^x+2e^x+C\Rightarrow I=e^x(x^2-2x+2)+C$