Question

Solve the given integral:$\int \frac{1}{\sqrt{1+\sin x}}dx$

Answer 1

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

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

$=\int \frac{1}{\sqrt{2}\sin \frac{\pi }{4}\sin \frac{x}{2}+\sqrt{2}\sin \frac{\pi }{4}\cos \frac{x}{2}}dx$

$=\int \frac{1}{\sqrt{2}\cos (\frac{x}{2}-\frac{\pi }{4})}dx$

$=\int \frac{1}{\sqrt{2}}\sec (\frac{x}{2}-\frac{\pi }{4})dx$

$=\frac{2}{\sqrt{2}}\ln (\sec (\frac{x}{2}-\frac{\pi }{4})-\tan (\frac{x}{2}-\frac{\pi }{4}))$

$=\sqrt{2}\ln (\sec (\frac{x}{2}-\frac{\pi }{4})-\tan (\frac{x}{2}-\frac{\pi }{4}))$