Q. $\int_{}^{}\frac{1}{1 + \cos{ax}}\ dx$ is equal to
A) $\cot{\frac{ax}{2} + c}$
B) $\frac{1}{a}\tan{\frac{ax}{2} + c}$
C) $\frac{1}{a}\ (cosec\ ax - \cot\ ax) + c$
D) $\frac{1}{a}(cosec\ ax + \cot{ax) + c}$
Explanation
(b)
$\int_{}^{}\frac{1}{1 + \cos{ax}}\ dx = \int_{}^{}\frac{1}{2\cos^{2}\left( \frac{ax}{2} \right)}\ dx$
$= \frac{1}{2}\ \int_{}^{}{\sec^{2}\frac{ax}{2}}\ dx = \frac{1}{2}.\ \frac{\tan\frac{\ \ \ ax}{2}}{\frac{a}{2}}$
$= \frac{1}{a}\tan{\frac{ax}{2} + c}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.