Intergration problem without using residue theorem

Question

I was wondering if anyone could evaluate the following integral, I know how to intergrate the function using the residue theorem , i was just wondering if there is a way to do it without the use of complex analysis

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

Answer 1

convert your Integrand into $$-{\frac {\cos \left( x \right) \left( 4\, \left( \cos \left( x \right) \right) ^{2}-3 \right) }{-5+4\,\cos \left( x \right) }} $$