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JEE Main & Advanced Mathematics Definite Integration Question Bank Summation of series by Definite Integration, Gamma function, Leibnitz's rule

  • question_answer
    If \[F(x)=\int_{{{x}^{2}}}^{{{x}^{3}}}{\log t\,dt,\,\,(x>0),}\] then \[{F}'(x)=\]        [MP PET 2001]

    A)                 \[(9{{x}^{2}}-4x)\log x\]               

    B)                 \[(4x-9{{x}^{2}})\log x\]

    C)                 \[(9{{x}^{2}}+4x)\log x\]              

    D)                 None of these

    Correct Answer: A

    Solution :

                       \[F(x)=\int_{{{x}^{2}}}^{{{x}^{3}}}{\log t\,dt}\]            Applying Leibnitz?s theorem,            \[F\,'(x)=\log {{x}^{3}}.\frac{d}{dx}{{x}^{3}}-\log {{x}^{2}}.\frac{d}{dx}{{x}^{2}}\]                          \[=3\log x.3{{x}^{2}}-2\log x.2x\] \[=(9{{x}^{2}}-4x)\log x\].


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