complex analysis - Fiding $int_0^{2pi}frac{mathrm{d}x}{4cos^2x+sin^2x}$

I was checking some old complex analysis homework and I found the following definite integral $$\int_0^{2\pi}\frac{\mathrm{d}x}{4\cos^2x+\sin^2x},$$
had to be found with the residue theorem. Back at the time I thought it was trivial, however I'm trying to do it, but I have no idea on how to star. Could anyone please give me a hint on how to start?

Thanks.

Answer

Hint: Let $z=e^{i x}$; $dx=-i dz/z$; $\cos{x}=(z+z^{-1})/2$. The integral is then equal to

$$-i \oint_{|z|=1} \frac{dz}{z} \frac{1}{1+\frac{3}{4} (z+z^{-1})^2} $$

Multiply out, determine the poles, figure out which poles, if any, lie within the unit circle, find the residues of those poles, multiply the sum of those residues (there may only be one, or none) by $i 2 \pi$, and you are done.