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JEE Main & Advanced Mathematics Definite Integration Question Bank Fundamental definite integration, Definite integration by substitution

  • question_answer
    The integral \[\int_{-1}^{3}{\left( {{\tan }^{-1}}\frac{x}{{{x}^{2}}+1}+{{\tan }^{-1}}\frac{{{x}^{2}}+1}{x} \right)}\,dx=\]                                                                 [Karnataka CET 2000]

    A)                 \[\pi \] 

    B)                 \[2\pi \]

    C)                 \[3\pi \]               

    D)                 None of these

    Correct Answer: B

    Solution :

               \[I=\int_{-1}^{3}{\left\{ {{\tan }^{-1}}\left( \frac{x}{{{x}^{2}}+1} \right)+{{\tan }^{-1}}\left( \frac{{{x}^{2}}+1}{x} \right) \right\}}dx\]                       \[=\int_{-1}^{3}{\left\{ {{\tan }^{-1}}\left( \frac{x}{{{x}^{2}}+1} \right)+{{\cot }^{-1}}\left( \frac{x}{{{x}^{2}}+1} \right) \right\}}dx\]                                    \[=\int_{-1}^{3}{\frac{\pi }{2}dx=}2\pi \].


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