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\int_{-\pi}^{\pi}\frac{1-r^2}{1-2r\cos\theta + r^2}d\theta & = & (2-2r^2)\int_{0}^{\pi}\frac{1}{1-2r\cos\theta + r^2}d\theta

\end{alignat}

We can do the above since the Poisson kernel is even. Wolfram says to make some trig subs which are easily doable but is there a way to integrate in another fashion.

We can use Complex Integration, Residue Theory, or other technique. I would never think of the substitution Wolfram gave so I would like to find a way to do this that is understandable.