Question

\(\textstyle \int_0^{\pi}x~log~sin~x~dx=\)

• A.
• B.
  
• C.
  
• D.
  

Answer 1

The correct option is B

$I={\int }_0^{\pi }xlogsinxdx\cdots \left(i\right)$
$={\int }_0^{\pi }(\pi -x)logsin(\pi -x)dx\cdots \left(ii\right)$
By adding (i) and (ii), we get
$2I={\int }_0^{\pi }\pi logsinxdx\Rightarrow I=\frac{2\pi }{2}{\int }_0^{\frac{\pi }{2}}logsinxdx$
$=\pi \left(\frac{\pi }{2}log\frac{1}{2}\right)=\frac{{\pi }^2}{2}log\frac{1}{2}$