We talked about two integrals, one $$ \int_{\theta=0}^{2\pi} \frac{1}{1 + \epsilon \cos(\theta)} d\theta. $$ The other is of the form $$ \int_{x=0}^\infty \frac{1}{1+x^n} dx $$

For the first integral, we replaced $\cos(\theta) = [e^{i\theta} + e^{-i\theta} ] / 2$, then replace $e^{i\theta} = z$, $d\theta = dz/(iz)$let $z$ run on the unit circle. We then get $$ \int_{|z|=1} \frac{1}{1 + \epsilon (z+1/z)/2} dz/(iz) $$