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

  • question_answer
    \[\int_{\,0}^{\,1}{\,{{\tan }^{-1}}\left( \frac{1}{{{x}^{2}}-x+1} \right)\,dx}\] is                                      [Orissa JEE 2003]

    A)                 ln 2        

    B)                 \[-\ln 2\]

    C)                 \[\frac{\pi }{2}+\ln 2\]   

    D)                 \[\frac{\pi }{2}-\ln 2\]

    Correct Answer: D

    Solution :

               \[\int_{0}^{1}{{{\tan }^{-1}}\left( \frac{1}{{{x}^{2}}-x+1} \right)\,dx}=\int_{0}^{1}{{{\tan }^{-1}}x\,dx-}\int_{0}^{1}{{{\tan }^{-1}}(x-1)}\,dx\]                                     \[=2\int_{\,0}^{\,1}{{{\tan }^{-1}}x\,dx}=2\,[{{\tan }^{-1}}x-\frac{1}{2}\log (1+{{x}^{2}})]_{0}^{1}=\frac{\pi }{2}-\log 2.\]


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