Question

If ${\int }_0^{\pi }{\Theta }^3\log \sin \Theta d\Theta =k{\int }_0^{\pi }{\Theta }^2\log \left(\sqrt{2}\sin \Theta \right)d\Theta$, then $k$ equals

• A. $\frac{\pi }{2}$
• B. $\pi$
• C. $\frac{3\pi }{2}$
• D. $2\pi$

Answer 1

The correct option is C $\frac{3\pi }{2}$

${\int }_0^{\pi }{\Theta }^3\log \sin \Theta d\Theta =\frac{1}{2}{\int }_0^{\pi }({\Theta }^3\log \sin \Theta +{(\pi -\Theta )}^3\log \sin (\pi -\Theta ))d\Theta$
$=\frac{1}{2}{\int }_0^{\pi }\left({\pi }^3\log \sin \Theta \right)d\Theta -\frac{3\pi }{2}{\int }_0^{\pi }\Theta (\pi -\Theta )\log \sin \Theta d\Theta$
$=\frac{3\pi }{2}{\int }_0^{\pi }{\Theta }^2\log \left(\sqrt{2}\sin \Theta \right)d\Theta$