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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
    Let\[\frac{d}{dx}F(x)=\left( \frac{{{e}^{\sin x}}}{x} \right)\,;\,x>0\]. If \[\int_{\,1}^{\,4}{\frac{3}{x}{{e}^{\sin {{x}^{3}}}}dx=F(k)-F(1)}\], then one of the possible value of k, is               [AIEEE 2003]

    A)                 15          

    B)                 16

    C)                 63          

    D)                 64

    Correct Answer: D

    Solution :

                       \[\frac{d}{dx}F(x)=\frac{{{e}^{\sin x}}}{x}\]\[\Rightarrow \int_{\,1}^{\,4}{\frac{3}{x}{{e}^{\sin {{x}^{3}}}}dx=\int_{\,1}^{\,4}{\frac{3{{x}^{2}}}{{{x}^{3}}}{{e}^{\sin {{x}^{3}}}}dx}}\]            Put \[{{x}^{3}}=t\,\,\,\Rightarrow 3\,{{x}^{2}}dx=dt\]            \[F(t)=\int_{1}^{64}{\frac{{{e}^{\sin t}}}{t}\,}dt=\int_{1}^{64}{F(t)dt=F(64)-F(1)},\]                                 On comparing, \[k=64.\]


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