Question

Integrate : $\int \frac{dx}{\sqrt{x-x^2}}$

Answer 1

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

Let $x={\sin }^2\theta \Rightarrow dx=2\sin \theta \cos \theta d\theta$

$I=\int \frac{2\sin \theta \cos \theta d\theta }{\sin \theta \cos \theta }$

$=\int 2d\theta =2\theta +c=2{\sin }^{-1}\left(\sqrt{x}\right)+c$

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