Q. $\int_{0}^{2\pi}{\sin^{6}{x\cos^{5}{x\ dx}}}$ is equal to
A) $2\pi$
B) $\pi/2$
C) 0
D) $- \pi$
Explanation
(c)
Let $I = \int_{0}^{2\pi}{sin^{6}xcos^{5}xdx}$
$= 2\int_{0}^{\pi}{\sin^{6}{x\cos^{5}{xdx}}}$ $\lbrack\because f(2\pi - x) = f(x)\rbrack$
Let $f(x) = \sin^{6}{x\cos^{5}x}$
$f(\pi - x) = \sin^{6}{(\pi - x)\cos^{5}{(\pi - x)}}$
$= - \sin^{6}x.\cos{\ ^{5}x = - f(x)}$
$\therefore\ I = 0$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.