Prove that \[ \int_{0}^{\pi / 2} \frac{\sqrt{\cot x}}{(1+\sqrt{\cot...

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Prove that
\[
\int_{0}^{\pi / 2} \frac{\sqrt{\cot x}}{(1+\sqrt{\cot x})} d x=\frac{\pi}{4}
\]
(IV)
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