question_answer

Evaluate \[\int{\frac{1}{\sqrt{1+\sin 2x}}}\,\,dx.\]

Answer:

Let \[l=\int{\frac{1}{\sqrt{1+\sin 2x}}\,dx}=\int{\frac{1}{\sqrt{1-\cos \left( \frac{\pi }{2}+2x \right)}}\,dx}\] \[=\int{\frac{1}{\sqrt{2{{\sin }^{2}}\left( \frac{\pi }{4}+x \right)}}\,dx}=\frac{1}{\sqrt{2}}\int{\text{cosec}\left( \frac{\pi }{4}+x \right)\,dx}\] \[=\frac{1}{\sqrt{2}}\log \left| \tan \left( \frac{\pi }{8}+\frac{x}{2} \right) \right|+C\]