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JEE Main & Advanced Mathematics Indefinite Integrals Question Bank Integration by Parts

  • question_answer
    \[\int_{{}}^{{}}{\frac{1}{{{\log }_{x}}e}dx=}\]      [MP PET 1994]

    A)                 \[\log {{\log }_{x}}e+c\]

    B)                 \[\frac{1}{{{({{\log }_{x}}e)}^{2}}}+c\]

    C)                 \[x\log \left( \frac{x}{e} \right)+c\]

    D)                 None of these

    Correct Answer: C

    Solution :

                    \[\int_{{}}^{{}}{\frac{1}{{{\log }_{x}}e}\,dx=\int_{{}}^{{}}{{{\log }_{e}}x\,dx}=x\log x-x+c}\]                                        \[=x({{\log }_{e}}x-{{\log }_{e}}e)+c=x{{\log }_{e}}\left( \frac{x}{e} \right)+c.\]


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