Definite Integration Question 450
Question: $ \int_0^{\pi }{x{{\sin }^{3}}xdx}= $
[CEE 1993]
Options:
A) $ \frac{4\pi }{3} $
B) $ \frac{2\pi }{3} $
C) 0
D) None of these
Show Answer
Answer:
Correct Answer: B
Solution:
Let $ I=\int_0^{\pi }{x{{\sin }^{3}}xdx} $ ……(i) Also $ I=\int_0^{\pi }{(\pi -x){{\sin }^{3}}xdx} $
…..(ii)
Adding (i) and (ii), we get $ 2I=\pi \int_0^{\pi }{{{\sin }^{3}}x}dx=\frac{\pi }{4}\int_0^{\pi }{{3\sin x-\sin 3x}dx} $
$ =\frac{\pi }{4}[ -3\cos x+\frac{\cos 3x}{3} ]_0^{\pi }=\frac{\pi }{4}[ 3-\frac{1}{3}+3-\frac{1}{3} ]=\frac{4\pi }{3} $
Hence, $ I=\frac{2\pi }{3} $ .
BETA
