Question

Evaluate :

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

Answer 1

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

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

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

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

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