Question

Evaluate: $\stackrel{1}{\underset{0}{\int }}{\sin }^{-1}\left(\frac{2x}{1+x^2}\right)dx$

Answer 1

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

$={\int }_0^{1}2ta{n}^{-1}xdx$

$=2ta{n}^{-1}x\int dx-\int (\frac{d}{dx}2ta{n}^{-1}\int dx)dx$

$=2ta{n}^{-1}x.x-\int \frac{2x}{1+x}dx$

$=2xta{n}^{-1}x-ln(1+x^2){|}_0^1$

$=2ta{n}^{-1}-ln2-(0-ln1)$

$=2\frac{\pi }{4}-ln2$

$=\frac{\pi }{2}-ln2$

$=\therefore {\int }_0^{1}si{n}^{-1}\left(\frac{2x}{1+x^2}\right)dx=\frac{\pi }{2}-ln2$