A) \[\log \tan \left( \frac{x}{2}+\frac{\pi }{2} \right)+c\]
B) \[\frac{1}{2}\log \tan \left( \frac{x}{2}+\frac{\pi }{6} \right)+c\]
C) \[\log \cot \left( \frac{x}{2}+\frac{\pi }{6} \right)+c\]
D) \[\frac{1}{2}\log \cot \left( \frac{x}{2}+\frac{\pi }{6} \right)+c\]
Correct Answer: B
Solution :
\[\int_{{}}^{{}}{\frac{dx}{\sin x+\sqrt{3}\cos x}}=\frac{1}{2}\int_{{}}^{{}}{\frac{dx}{\frac{\sin x}{2}+\frac{\sqrt{3}}{2}\cos x}}\] \[=\frac{1}{2}\int_{{}}^{{}}{\frac{dx}{\sin \left( x+\frac{\pi }{3} \right)}}=\frac{1}{2}\int_{{}}^{{}}{\text{cosec}\left( x+\frac{\pi }{3} \right)}\] \[=\frac{1}{2}\log \tan \left( \frac{x}{2}+\frac{\pi }{6} \right)+c.\]
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