Solve complex integral: $\int_{0}^{2\pi} \frac{\cos{3\theta}}{5-4\cos\theta}d\theta$

Question

I'm trying to show that the following complex integral is:

$$\int_{0}^{2\pi} \frac{\cos{3\theta}}{5-4\cos\theta}d\theta = \frac{\pi}{12}$$

I have thought about calculating the residues of this function at where the denominator becomes zero but as $\cos\theta$ is always less or equal to 1, I don't know how to apply this method.

Is there any other way for solving this kind of integrals? Please, could you give me any hint?

Thanks!

Answer 1

Hint:

$$\int_{0}^{2\pi} \frac{\cos3\theta}{5 - 4\cos\theta} \ d\theta = \frac{i}{2}\int_{C} \frac{z^3 +z^{-3}}{z(5 - 2(z+z^{-1}))} \ dz $$

Where $C$ is the unit circle and $z = e^{i\theta}$. Simplifying gives

$$\frac{1}{2i}\int_C\frac{z^6+1}{z^3(-2z^2+5z -2)} $$

Then apply the residue theorem.