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
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