A) \[\frac{1}{2}\log \,\tan \,\left( \frac{x}{2}+\frac{\pi }{12} \right)+C\]
B) \[\frac{1}{2}\log \tan \,\left( \frac{x}{2}-\,\frac{\pi }{12} \right)\,+C\]
C) \[\log \,\tan \,\left( \frac{x}{2}+\frac{\pi }{12}\, \right)+C\]
D) \[\log \,\tan \,\left( \frac{x}{2}-\frac{\pi }{12} \right)+C\]
Correct Answer: A
Solution :
\[\int{\frac{dx}{\cos x+\sqrt{3}\sin x}}\] \[=\int{\frac{dx}{2\left( \frac{1}{2}\cos x+\frac{\sqrt{3}}{2}\sin x \right)}}\] \[=\frac{1}{2}\int{\sec \left( x-\frac{\pi }{3} \right)}dx\] \[=\frac{1}{2}\log \,\tan \left( \frac{x}{2}-\frac{\pi }{6}+\frac{\pi }{4} \right)+C\] \[=\frac{1}{2}\log \,\tan \left( \frac{x}{2}+\frac{\pi }{12} \right)+C\]
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