Evaluate the following integrals:

Let $I=\int x e^{2 x} d x$

Using integration by parts,

$I=x \int e^{2 x} d x-\int \frac{d}{d x} x \int e^{2 x} d x$

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

$=\frac{x e^{2 x}}{2}-\int \frac{e^{2 x}}{2} d x$

$=\frac{x e^{2 x}}{2}-\frac{e^{2 x}}{4}+c$

$I=\left(\frac{x}{2}-\frac{1}{4}\right) e^{2 x}+c$