Question

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

Answer 1

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

Put $\sin x-\cos x=t$

$\left(\sin x+\cos x\right)dx=dt$

$t^2=1-\sin 2x$

$\Rightarrow \sin 2x=1-t^2$

$=\int \frac{dt}{\sqrt{1-t^2}}$

$={\sin }^{-1}t+c$

$={\sin }^{-1}\left(\sin x-\cos x\right)$

$={\sin }^{-1}\left(\sqrt{2}\sin \left(x-\frac{\pi }{4}\right)\right)tc$