Calculating $\int_0^{2\pi} \exp(int)\exp(-imt)dt$ , $\int_0^{2\pi} \sin(nt)\cos(mt)dt$

Question

What's the easiest/most efficient way to calculate integrals like:

$\int_0^{2\pi} \exp(int)\exp(-imt)dt$

$\int_0^{2\pi} \sin(nt)\cos(mt)dt$

I know that $\exp(int)=\cos(nt)+i\sin(nt)$ and $\sin(nt)=\frac{\exp(int)-\exp(-int)}{2i}$ and $\cos(mt)=\frac{\exp(imt)+\exp(-imt)}{2}$.

Answer 1

For the first, use $$e^a.e^b=e^{a+b} $$

for the second

$$2\sin (a)\cos (b)=\sin(a+b)+\sin (a-b) $$