A) \[\cot \,\frac{ax}{2}+c\]
B) \[\frac{1}{a}\,\tan \,\frac{ax}{2}+c\]
C) \[\frac{1}{a}(\text{cosec ax - cot ax)+c}\]
D) \[\frac{1}{a}(\text{cosec ax + cot ax)+c}\]
Correct Answer: B
Solution :
\[\int{\frac{1}{1+\cos \,ax}}\,dx\] \[=\int{\frac{dx}{2\,{{\cos }^{2}}\,(ax/2)}}\] \[=\frac{1}{2}\int{{{\sec }^{2}}\,\frac{ax}{2}\,dx=\frac{1}{2}}.\frac{\tan ax/2}{a/2}\] \[=\frac{1}{a}\tan \frac{ax}{2}+c\]
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