Evaluate the following integrals:

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

$=\frac{1}{2} \int \mathrm{e}^{2 \mathrm{x}} 2 \cos ^{2} \mathrm{x} \mathrm{dx}$

$=\frac{1}{2} \int \mathrm{e}^{2 \mathrm{x}}(1+\cos 2 \mathrm{x}) \mathrm{dx}$

$=\frac{1}{2} \int \mathrm{e}^{2 \mathrm{x}} \mathrm{dx}+\frac{1}{2} \int \mathrm{e}^{2 \mathrm{x}} \cos 2 \mathrm{xdx}$

We know that, $\int e^{a x} \cos b x d x=\frac{e^{a x}}{a^{2}+b^{2}}\{a \cos b x-b \sin b x\}+c$

$I=\frac{1}{2}\left[\frac{e^{2 x}}{2}-\frac{e^{2 x}}{8}(2 \cos 2 x+2 \sin 2 x)\right]+c$

$=\frac{e^{2 x}}{4}+\frac{e^{2 x}}{16}(2 \cos 2 x+2 \sin 2 x)+c$

$=\frac{e^{2 x}}{4}+\frac{e^{2 x}}{8}(\cos 2 x+\sin 2 x)+c$