Integrating $\int_0^{2\pi}\frac{1}{1+8\cos^2\theta}d\theta$ Can someone please help me integrate $$\int_0^{2\pi}\frac{1}{1+8\cos^2\theta}d\theta$$ the question says, as a hint, use $\cos\theta = \frac{z + z^{-1}}{2}$ with $|z|=1$. I'm not really sure where to start. 

An easy way to evaluate:   
$I=\int_0^{2\pi}\frac{1}{1+8\cos^2\theta}d\theta$
$$I=4\int_0^{\frac{\pi}{2}}\frac{1}{8+\frac{1}{\cos^2\theta}}\frac{d\theta}{\cos^2\theta }= 4\int_0^{\frac{\pi}{2}}\frac{1}{9+\tan^2\theta}d\tan\theta= 4\int_0^{\infty}\frac{1}{3^2+t^2}dt= 2\pi/3$$
