Q. $\int_{- \pi/3}^{\pi/3}{\frac{x\sin x}{\cos^{2}x}dx}$ is equal to
A) $\frac{1}{3}(4\pi + 1)$
B) $\frac{4\pi}{3} - 2\log{\tan\frac{5\pi}{12}}$
C) $\frac{4\pi}{3} + \log{\tan\frac{5\pi}{12}}$
D) None of these
Explanation
(b)
Let $I = \int_{- \pi/3}^{\pi/3}{x\sec{x\tan{x\ dx\ \ \lbrack x\sec{x\tan\ x\ }is\ an\ even\rbrack}}}$
$= 2\int_{0}^{\pi/3}{x\sec{x\tan{x\ dx}}}$
$= 2{\left\lbrack x\sec{x - \log\left\{ \tan\left( \frac{\pi}{4} + \frac{x}{2} \right) \right\}} \right\rbrack_{0}}^{\pi/3}$
$= \frac{4\pi}{3} - 2\log{\tan\frac{5\pi}{12}}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.