A) \[2\sqrt{2}\log \tan \left( \frac{\pi }{8}+\frac{x}{4} \right)+c\]
B) \[\frac{1}{\sqrt{2}}\log \tan \left( \frac{\pi }{8}+\frac{x}{4} \right)+c\]
C) \[\sqrt{2}\log \tan \left( \frac{\pi }{8}+\frac{x}{4} \right)+c\]
D) \[\frac{1}{2\sqrt{2}}\log \tan \left( \frac{\pi }{8}+\frac{x}{4} \right)+c\]
Correct Answer: C
Solution :
\[\int_{{}}^{{}}{\frac{1}{\sqrt{1+\sin x}}}\,dx=\int_{{}}^{{}}{\frac{1}{\sqrt{2}\sin \left( \frac{\pi }{4}+\frac{x}{2} \right)}}\,dx\] \[=\frac{1}{\sqrt{2}}\int_{{}}^{{}}{\text{cosec}\,\left( \frac{x}{\text{2}}+\frac{\pi }{4} \right)}\,dx=\sqrt{2}\log \tan \left( \frac{\pi }{8}+\frac{x}{4} \right)+c.\]
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