CVRC7

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Prove that \int _{0}^{{\pi /2}}{\frac  {r\theta \sin(2\theta )}{1-2r\cos(2\theta )+r^{2}}}\ d\theta ={\begin{cases}{\frac  {\pi }{4}}\ln(1+r)&{\mathrm  {if}}\ r^{2}<1\\{\frac  {\pi }{4}}\ln(1+1/r)&{\mathrm  {if}}\ r^{2}>1\end{cases}}.   Hint: Integrate {\frac  {2zr}{z^{2}(1+r)^{2}+(1-r)^{2}}}{\frac  {{\mathrm  {Log}}(1-iz)}{1+z^{2}}} over the semicircle contour in the upper half plane, then put x=\tan \theta \,.