I have an integral to evaluate via contour integration:

$\displaystyle ({1 \over 1+cos^2\theta}) \ wrt d\theta$

Using the substitution $\displaystyle z =e^{i\theta}$I find the integral becomes,with $\displaystyle d\theta \ becoming \ {dz \over iz}$

$\displaystyle {4z^2 \over z^4 + 6z^2 + 1}$ is that correct?

Also how would I evaluate this via the residue theorem?