Question

$\int \frac{\cos \left({\tan }^{-1}x\right)}{\left(1+x^2\right)\sqrt{\sin \left({\tan }^{-1}x\right)}}dx$

Answer 1

Consider the given integral.

$I=\int \frac{\cos \left({\tan }^{-1}x\right)}{(1+x^2)\sqrt{\sin \left({\tan }^{-1}x\right)}}dx$

Let $t={\tan }^{-1}x$

$dt=\left(\frac{1}{1+x^2}\right)dx$

Therefore,

$I=\int \frac{\cos t}{\sqrt{\sin t}}dt$

Let $p=\sin t$

$dp=\cos tdt$

Therefore,

$I=\int \frac{1}{\sqrt{p}}dp$

$I=2\sqrt{p}+C$

On putting the all values, we get

$I=2\sqrt{\sin t}+C$

$I=2\sqrt{\sin \left({\tan }^{-1}x\right)}+C$

Hence, this is the answer.