Mathematics

# Integrate $\int _{ -\frac { \pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ \log { \left( \dfrac { 2-\sin { \theta } }{ 2+\sin { \theta } } \right) } } d\theta =$

##### SOLUTION
$\int\limits_{ - \frac{\pi }{2}}^{\frac{\pi }{2}} {\log \left( {\dfrac{{2 - \sin \theta }}{{2 + \sin \theta }}} \right)d\theta }$
Let $f\left( \theta \right) = \log \dfrac{{2 - \sin \theta }}{{2 + \sin \theta }}$

$f\left( { - \theta } \right) = \log \dfrac{{2 - \sin \left( { - \theta } \right)}}{{2 + \sin \left( { - \theta } \right)}}$

$= \log \dfrac{{2 + \sin \theta }}{{2 - \sin \theta }}$

$= - \log \dfrac{{2 - \sin \theta }}{{2 + \sin \theta }}$

$= - f\left( \theta \right)$
So, $f\left( \theta \right)$ is an odd function , therefore $\int\limits_{ - \frac{\pi }{2}}^{\frac{\pi }{2}} {\log \left( {\dfrac{{2 - \sin \theta }}{{2 + \sin \theta }}} \right)d\theta } = 0$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
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