# CVRC7

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