$\displaystyle I=\int^{2\pi}_{0}\frac{cos(3\theta)}{5-4\cos(\theta)}d\theta$

$\displaystyle z=e^{i\theta},\frac{dz}{d\theta}=i\\e^{i\theta}=i\ \z\Rightarrow\\d\theta=\frac{dz}{i\\z}$

$\displaystyle I=\oint_{C_{1}(0)}\frac{z^{3}+z^{-3}}{2\\i(-2z^{2}+5z-2)}dz$

factorizing gives

$\displaystyle I=\oint_{C_{1}(0)}\frac{z^{3}+z^{-3}}{2\\i(z-2)(z-\frac{1}{2})}dz$

now from here.. i need to do $\displaystyle 2\pi\\i*\sum\\Res\\|_{z}$ but everytime i do it, i get dumb answers, im pretty sure the answer is $\displaystyle \frac{\pi}{12}$

please can anyone help me.. i may have done something stupid