Question

Solve : $\int \frac{1}{(1-x)\sqrt{x}}dx$ ?

Answer 1

$I=\int \frac{1}{(1-x)\sqrt{x}}dx=-\int \frac{1}{(x-1)\sqrt{x}}dxLet,x=u\Rightarrow u=\sqrt{x}\Rightarrow dx=2\sqrt{x}du\therefore I=-2\int \frac{1}{u^2-1}dunowbypartialfraction,I=\int (\frac{1}{u+1}-\frac{1}{u-1})du=log(u+1)-log(u-1)=log\left(\frac{u+1}{u-1}\right)+C=log\left(\frac{\sqrt{x}+1}{\sqrt{x-1}}\right)+C$