Question

Evaluate : $\int \frac{\sin x-\cos x}{\sqrt{\sin 2x}}dx$

Answer 1

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

Put $\cos x+\sin x=t(-\sin x+\cos x)dx=dt(\sin x-\cos x)-dt$

$I=\int \frac{-dt}{\sqrt{t^2-1}}=-ln|\ast +\sqrt{t^2-1}|+c=-ln|\cos x+\sin x+\sqrt{(\cos x+\sin x{)}^2-1}|I=-ln|\cos x+\sin x+\sqrt{\sin 2x}|+c$