Question

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

Answer 1

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

$\int \frac{\sin x-\cos x}{\sqrt{1+2\sin x.\cos x-1}}dx$

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

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

$=-\log |(\sin x+\cos x)+\sqrt{(\sin x+\cos x{)}^2-1}|+c$ [ $c$ being integrating constant]

$=-\log |(\sin x+\cos x)+\sqrt{\sin 2x}|+c$