Question

Evaluate $\int \frac{\cot x}{\sqrt{\sin x}}dx$

Answer 1

$I=\int \frac{\cot x}{\sqrt{\sin x}}dx$
$I=\int \frac{\cos x}{\sin x\sqrt{\sin x}}dx=\int \frac{\cos x}{(\sin x{)}^{3/2}}dx$
Let $t=\sin x$, then $dt=\cos xdx$
$\therefore I=\int \frac{1}{t^{3/2}}dt=\frac{t^{-1/2}}{-1/2}+C$
$I=-\frac{2}{\sqrt{t}}+C$
$\therefore I=-\frac{2}{\sqrt{\sin x}}+C$