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

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

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