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${\pi ^2}\log \frac{1}{2}$
$= \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}$.
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.