Computing an Integral Using Complex Analysis Does anyone know how to compute

$$
\int^{2\pi}_{0} \frac{1}{3 + \cos\theta} \, d\theta
$$

using tools from complex analysis? I'm not sure how to get started, or if complex analysis is even needed to do this.
Recall that $\cos\theta = (e^{i\theta}+e^{-i\theta})/2$, and change variables to $z=e^{i\theta}$.  Then you are integrating around a circle, and can look for poles.
Let $\cos \theta = \dfrac{z^2+1}{2z}$, $\,d\theta=\dfrac{dz}{iz}$.