Evaluate the following integrals:

Let $I=\int e^{2 x}(-\sin x+2 \cos x) d x$

$I=\int e^{2 x}-\sin x d x+2 \int e^{2 x} \cos x d x$

Applying by parts in the second integral,

$I=-\int e^{2 x} \sin x d x+2\left\{\frac{1}{2} e^{2 x} \cos x+\int \frac{1}{2} e^{2 x} \sin x d x\right\}$

$=-\int e^{2 x} \sin x d x+e^{2 x} \cos x+\int e^{2 x} \sin x d x+c$

$=e^{2 x} \cos x+c$