A) \[\log \,tan\left( \frac{x}{2}+\frac{\pi }{12} \right)+C\]
B) \[\log \,tan\left( \frac{x}{2}-\frac{\pi }{12} \right)+C\]
C) \[\frac{1}{2}\log \,tan\left( \frac{x}{2}+\frac{\pi }{12} \right)+C\]
D) \[\frac{1}{2}\log \,tan\left( \frac{x}{2}-\frac{\pi }{12} \right)+C\]
Correct Answer: C
Solution :
| \[I=\int{\frac{dx}{\cos x+\sqrt{3}\sin x}}\] |
| \[I=\int{\frac{dx}{2\left[ \frac{1}{2}\cos x+\frac{\sqrt{3}}{2}\sin x \right]}}\]\[=\frac{1}{2}\int{\frac{dx}{\left[ \sin \frac{\pi }{6}\cos x+\cos \frac{\pi }{6}\sin x \right]}}=\frac{1}{2}\int{\frac{dx}{\sin \left( x+\frac{\pi }{6} \right)}}\] |
| \[\Rightarrow \,\,I=\frac{1}{2}\int{\text{cosec }\,\left( x+\frac{\pi }{6} \right)dx}\] |
| \[\because \,\,\int{\text{cosec x dx=log }\!\!|\!\!\text{ (tan x/2) }\!\!|\!\!\text{ }}+C\] |
| \[\therefore \,\,I=\frac{1}{2}\log \,\tan \left( \frac{x}{2}+\frac{\pi }{12} \right)+C\] |
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