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