Question

Integrate ${\int }_0^1{\sin }^{-1}\left(\frac{2x}{1+x^2}\right)dx$

Answer 1

Let,

$I={\int }_0^{1}si{n}^{-1}\left(\frac{2x}{1+x^2}\right)dx$

Let $x=tan\theta$

$dx=se{c}^2\theta d\theta$

When, $x=0,\theta =0$ & $x=1,\theta =\frac{\pi }{4}$

Now,

$I={\int }_0^{\frac{\pi }{4}}2\theta .{\sec }^2\theta .d\theta$

Integrating by parts , Taking $se{c}^2\theta$ as a first function.

$=\frac{\pi }{2}-\log 2$