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JEE Main & Advanced Mathematics Definite Integration Question Bank Properties of Definite Integration

  • question_answer
    \[\int_{-\pi /2}^{\pi /2}{\log \left( \frac{2-\sin \theta }{2+\sin \theta } \right)\,d\theta =}\]

    A)                 0             

    B)                 1

    C)                 2             

    D)                 None of these

    Correct Answer: A

    Solution :

               Since \[f(-\theta )=\log {{\left( \frac{2-\sin \theta }{2+\sin \theta } \right)}^{-1}}=-\log \left( \frac{2-\sin \theta }{2+\sin \theta } \right)=-f(\theta )\]                    \[\therefore \] \[f(x)\] is an odd function of \[x\].                                 Therefore, \[2\int_{0}^{\pi /2}{\log \left( \frac{2-\sin \theta }{2+\sin \theta } \right)\text{ }d\theta =0}\].


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