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JEE Main & Advanced Mathematics Definite Integration Question Bank Properties of Definite Integration

  • question_answer
    For any integer \[n,\] the integral                 \[\int_{0}^{\pi }{{{e}^{{{\sin }^{2}}x}}{{\cos }^{3}}(2n+1)x\,dx=}\]                                            [MNR 1982]

    A)                 \[-1\]    

    B)                 0

    C)                 1             

    D)                 \[\pi \]

    Correct Answer: B

    Solution :

               Let \[f(x)=\int_{0}^{\pi }{{{e}^{{{\sin }^{2}}x}}{{\cos }^{3}}(2n+1)x.dx}\]            Since \[\cos (2n+1)(\pi -x)=\cos [(2n+1)\pi -(2n+1)x]\]            \[=-\cos (2n+1)x\]and \[{{\sin }^{2}}(\pi -x)={{\sin }^{2}}x\]            Hence by the property of definite integral,                                 \[\int_{0}^{\pi }{{{e}^{{{\sin }^{2}}x}}{{\cos }^{3}}(2n+1)x\,dx=0}\],  \[[f(2a-x)=-f(x)]\].


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