_0^ x ^3 x ,dx = - Vidyadip

MCQ

$\int_0^\pi {x{{\sin }^3}x\,dx} = $

A
$\frac{{4\pi }}{3}$
✓
$\frac{{2\pi }}{3}$
C
$0$
D
None of these

✓

Answer

Correct option: B.

$\frac{{2\pi }}{3}$

b
(b) Let $I = \int_0^\pi {x{{\sin }^3}x\,dx} $....$(i)$

Also $I = \int_0^\pi {(\pi - x){{\sin }^3}x\,\,dx} $.....$(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}\left[ { - 3\cos x + \frac{{\cos 3x}}{3}} \right]_0^\pi $

$= \frac{\pi }{4}\left[ {3 - \frac{1}{3} + 3 - \frac{1}{3}} \right] = \frac{{4\pi }}{3}$

Hence, $I = \frac{{2\pi }}{3}$.

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