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