Question

Evaluate $\int \frac{1}{\sqrt{1+\sin 2x}}dx$

Answer 1

$I=\int \frac{1}{\sqrt{1+\sin 2x}}dx$

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

$I=\int \frac{1}{\sqrt{2{\sin }^2(\frac{\pi }{4}+x)}}dx$

$I=\frac{1}{\sqrt{2}}\int cosec(\frac{\pi }{4}+x)dx$

$I=\frac{1}{\sqrt{2}}\log |\tan (\frac{\pi }{8}+\frac{x}{2})|+C$