How do I compute $\int_{0}^{2\pi}\frac{1}{a+b\cos x}dx$ using contour integration?

Question

I'm trying to compute $\int_{0}^{2\pi}\frac{1}{a+b\cos x}dx$ with $a,b$ being positive reals. What function should I choose which has as real part this function on the real axis? And what contour should I take?

Answer 1

Hint. Note that $2\cos(x)=z+\frac{1}{z}$ with $z=e^{ix}$. What is the set $\{e^{ix} : x\in [0,2\pi]\}?$

You should also pay attention to how the positive real numbers $a$ and $b$ are chosen.

What happens when $a>b$? What about $a\leq b$?