Why is this integral equal to $\frac{2\sinh(\pi)}{(1+n^2)\pi}$

Question

It is true that $$\frac{(-1)^n}{\pi}\int\limits_{-\pi}^{\pi}e^y\cos(ny)dy=\frac{2\sinh(\pi)}{(1+n^2)\pi}$$ but I don't see why. Could someone write some intermediate steps please?

Answer 1

Note that $$\int\limits_{-\pi}^{\pi}e^y\cos(ny)dy=\text{Re}\left(\int\limits_{-\pi}^{\pi}e^{(1+in)y}dy\right)= \text{Re}\left(\frac{\left[e^{(1+in)y}\right]_{-\pi}^{\pi}}{1+in}\right).$$ Can you take it from here?