Definite Integration Question 22
Question: $ \int _{0}^{\pi }{\log {{\sin }^{2}}xdx=} $
[MP PET 1997]
Options:
A) $ 2\pi {\log _{e}}( \frac{1}{2} ) $
B) $ \pi {\log _{e}}2+c $
C) $ \frac{\pi }{2}{\log _{e}}( \frac{1}{2} )+c $
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
$ \int_0^{\pi }{2\log \sin xdx=2\int_0^{2\frac{\pi }{2}}{\log \sin xdx=4\int_0^{\pi /2}{\log \sin xdx}}} $
$ =4\times ( -\frac{\pi }{2}\log 2 )=-2\pi {\log _{e}}2=2\pi {\log _{e}}( \frac{1}{2} ) $ .
BETA
