Question

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

Answer 1

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

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

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

$=\frac{1}{\sqrt{2}}\int (\sqrt{\tan x}+\sqrt{\cot x})dx$

$=\frac{1}{\sqrt{2}}[\frac{(\tan x{)}^{\frac{3}{2}}}{\frac{3}{2}}\log \sec x+\frac{(\cot x{)}^{\frac{3}{2}}}{\frac{3}{2}}\log \sin x]$

$=\frac{\sqrt{2}}{3}\left[\right(\tan x{)}^{\frac{3}{2}}\log \sec x+\left(\cot x{)}^{\frac{3}{2}}\log \sin x\right]+C$