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

  • question_answer
    The value of \[\int_{0}^{\pi }{{{e}^{{{\cos }^{2}}x}}{{\cos }^{5}}3x}\,dx\] is           [Bihar CEE 1994]

    A)                 1             

    B)                 \[-1\]

    C)                 0             

    D)                 None of these

    Correct Answer: C

    Solution :

       Let\[I=\int_{0}^{\pi }{{{e}^{{{\cos }^{2}}x}}{{\cos }^{5}}3x\,dx=\int_{0}^{\pi }{{{e}^{{{\cos }^{2}}x}}{{\cos }^{5}}3(\pi -x)dx}}\],                                 \[[\because \int_{0}^{a}{f(x)dx=}\int_{0}^{a}{f(a-x)dx]}\]                                      \[I=-\int_{0}^{\pi }{{{e}^{{{\cos }^{2}}x}}{{\cos }^{5}}3xdx=-I}\]Þ \[2I=0\Rightarrow I=0\].


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