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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
    \[\int_{0}^{\pi /2}{{{\sin }^{4}}x{{\cos }^{6}}x\,dx}\] equals                                         [RPET 1999]

    A)                 \[\frac{5\pi }{512}\]       

    B)                 \[\frac{3\pi }{512}\]

    C)                 \[\frac{\pi }{512}\]          

    D)                 None of these

    Correct Answer: B

    Solution :

                       \[I=\int_{0}^{\pi /2}{{{\sin }^{4}}x{{\cos }^{6}}x.dx}\]            \[\Rightarrow I=\frac{\Gamma \,(5/2)\,\Gamma \,(7/2)}{2\Gamma (6)}\],  (Applying gamma function)                                 Þ \[I=\frac{3/2.1/2.\sqrt{\pi }.5/2.3/2.1/2.\sqrt{\pi }}{2.5.4.3.2.1}\]\[=\frac{3\pi }{512}\].


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