A) \[\log \tan \left( \frac{\pi }{8}+\frac{x}{2} \right)+c\]
B) \[\log \tan \left( \frac{\pi }{8}-\frac{x}{2} \right)+c\]
C) \[\frac{1}{\sqrt{2}}\log \tan \left( \frac{\pi }{8}+\frac{x}{2} \right)+c\]
D) None of these
Correct Answer: C
Solution :
\[\int_{{}}^{{}}{\frac{dx}{\sin x+\cos x}}=\frac{1}{\sqrt{2}}\int_{{}}^{{}}{\frac{dx}{\sin x\cos \frac{\pi }{4}+\cos x\sin \frac{\pi }{4}}}\] \[=\frac{1}{\sqrt{2}}\int_{{}}^{{}}{\text{cosec }\left( x+\frac{\pi }{4} \right)\,dx=\frac{1}{\sqrt{2}}\log \tan \left( \frac{\pi }{8}+\frac{x}{2} \right)}+c.\]
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