Question

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

Answer 1

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

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

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

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

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