# How to solve $\int_{0}^{2\pi} \frac{\cos(50x)}{5+4\cos(x)} dx\,?$

I encountered this integral and tried to solve it. As you can expect I could not solve this and thought I will ask it here.

The integral is:

$$\int_{0}^{2\pi} \frac{\cos(50x)}{5+4\cos(x)}\, dx$$

I don't know a way or I know it but I can't see which way or method I have to use.

If you know it then please help. If I see the technique once I will understand it.

You may use this formula: $$\int_0^{2\pi}\frac{\cos mx}{p-q\cos x}\, dx=\frac{2\pi}{\sqrt{p^2-q^2}}\left(\frac{p-\sqrt{p^2-q^2}}{q}\right)^m\qquad\hbox{for}\qquad |q|<p$$ The complete proof can be seen here.
Your integral can be evaluated by setting $m=50$, $p=5$, and $q=-4$.