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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
    The derivative of \[F(x)=\int_{{{x}^{2}}}^{{{x}^{3}}}{\frac{1}{\log t}\,dt}\], \[(x>0)\] is

    A)                 \[\frac{1}{3\log x}-\frac{1}{2\log x}\]     

    B)                 \[\frac{1}{3\log x}\]

    C)                 \[\frac{3{{x}^{2}}}{3\log x}\]      

    D)                 \[{{(\log x)}^{-1}}.x(x-1)\]

    Correct Answer: D

    Solution :

                       We know that \[\frac{d}{dx}\left( \int_{a}^{b}{f(t)dt} \right)=\frac{db}{dx}f(b)-\frac{da}{dx}f(a)\]                    a and b are functions of x.            \[\therefore \,\,\,F(x)=\int_{{{x}^{2}}}^{{{x}^{3}}}{\frac{1}{\log t}dt}\]            Þ \[F'(x)=\frac{d}{dx}({{x}^{3}})\frac{1}{\log {{x}^{3}}}-\frac{d}{dx}({{x}^{2}})\frac{1}{\log {{x}^{2}}}\]                                              \[=\frac{3{{x}^{2}}}{3\log x}-\frac{2x}{2\log x}=x(x-1){{(\log x)}^{-1}}\].


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