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MCQ

$\int_{\,0}^{\,\pi } {\log {{\sin }^2}x\,dx = } $

✓
$2\pi {\log _e}\left( {\frac{1}{2}} \right)$
B
$\pi {\log _e}2 + c$
C
$\frac{\pi }{2}{\log _e}\left( {\frac{1}{2}} \right) + c$
D
None of these

✓

Answer

Correct option: A.

$2\pi {\log _e}\left( {\frac{1}{2}} \right)$

a
(a) $\int_0^\pi {2\log \sin xdx = 2\int_0^{2\frac{\pi }{2}} {\log \sin xdx = 4\int_0^{\pi /2} {\log \sin x\,dx} } } $

$ = 4 \times \left( { - \frac{\pi }{2}\log 2} \right) = - 2\pi {\log _e}2 = 2\pi {\log _e}\left( {\frac{1}{2}} \right)$.

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