Question

Solve :
${\int }_0^1\frac{{\sin }^{-1}\sqrt{x}}{\sqrt{x(1-x)}}dx$

Answer 1

$si{n}^{-1}\sqrt{x}=t\Rightarrow \frac{1}{2\sqrt{x}\sqrt{1-x}}dx=dt$

$\therefore I={\int }_0^{\frac{\pi }{2}}t\left(2dt\right)=[t^2{]}_0^{\frac{\pi }{2}}$

$=\frac{{\pi }^2}{4}$