
Complex Integral
Show that
$\displaystyle \frac{1}{2 \pi} \int_{\pi}^{\pi} \frac{1}{12r\cos \theta +r^2} d \theta = 1, 0 \leq r < 1$.
Right now, I don't see how to show this. I do see that the integrand is an even function if that helps at all. Also, since this is not from $\displaystyle \infty$ to $\displaystyle \infty$, I do not know the method to proceed. I need a few hints on how to start.

Hi. If I use the $\displaystyle z=e^{it}$ substitution, then I get:
$\displaystyle \int_{\pi}^{\pi}\frac{1}{12 r\cos(t)+r^2}dt=i\oint \frac{dz}{(rz)(rz1)}=\frac{2\pi}{1r^2},\quad r<1$