_0^x x ,dx = - Vidyadip

MCQ

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

A
$\frac{\pi }{2}\log \frac{1}{2}$
✓
$\frac{{{\pi ^2}}}{2}\log \frac{1}{2}$
C
$\pi \log \frac{1}{2}$
D
${\pi ^2}\log \frac{1}{2}$

✓

Answer

Correct option: B.

$\frac{{{\pi ^2}}}{2}\log \frac{1}{2}$

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

$= \int_0^\pi {(\pi - x)\log \sin (\pi - x)\,dx} $.....$(ii)$

By adding $(i)$ and $(ii),$ we get

$2I = \int_0^\pi \pi \log \sin x\,dx $

$\Rightarrow I = \frac{{2\pi }}{2}\int_0^{\pi /2} {\log \sin \,x\,dx} $

$ = \pi \left( {\frac{\pi }{2}\log \frac{1}{2}} \right) = \frac{{{\pi ^2}}}{2}\log \frac{1}{2}$.

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