Evaluate the following integrals:

Let $I=\int x e^{x} d x$

Using integration by parts,

$I=x \int e^{x} d x-\int \frac{d}{d x} x \int e^{x} d x$

We know that, $\int e^{x} d x=e^{x}$ and $\frac{d}{d x} x=1$

$=x e^{x}-\int e^{x} d x$

$=x e^{x}-e^{x}+c$