Q. The value of integral $\int_{0}^{\pi}{x\ f(\sin{x)\ dx\ is}}$
A) 0
B) $\pi\int_{0}^{\pi/2}{f\ (\sin{x)\ dx}}$
C) $\frac{\pi}{4}\int_{0}^{\pi}{f\ (\sin{x)\ dx}}$
D) None of these
Explanation
(b)
Let $I = \int_{0}^{\pi}{x\ f(\sin{x)dx}}$
$\Rightarrow I = \int_{0}^{\pi}{(\pi - x)f\{\sin{(\pi - x)}\} dx}$
$= \pi\int_{0}^{\pi}{f(\sin{x)dx - x\int_{0}^{\pi}{f(\sin{x)dx}}}}$
$\Rightarrow I = \pi\int_{0}^{\pi}{f(\sin{x)dx - I}}$
$\Rightarrow 2I = 2\pi\int_{0}^{\pi/2}{f(\sin{x)dx}}$
$\Rightarrow I = \pi\int_{0}^{\pi/2}{f(\sin{x)dx}}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.