$$\int\limits_0^{\pi} \frac{\cos(mx)}{1 - a \cdot\cos x}\,dx, m \in \mathbb{N}, |a| < 1$$
I know that $$\frac{1 - a \cdot\cos(bx)}{1 - 2a\cdot\cos(bx) + a^2} = 1 + \sum_{n=1}^{\infty} a^n\cdot\cos(nbx), |a| < 1$$
Does it help to calculate this integral? Can you give me a hint?