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

  • question_answer
    \[\int_{\,0}^{\,\pi }{{{e}^{{{\sin }^{2}}x}}{{\cos }^{3}}x\,dx}\] is equals to                              [MP PET 2002]

    A)                 \[-1\]    

    B)                 0

    C)                 1             

    D)                 \[\pi \]

    Correct Answer: B

    Solution :

               \[I=\int_{0}^{\pi }{{{e}^{{{\sin }^{2}}x}}{{\cos }^{3}}x\,\,dx}\]                    \[\Rightarrow I=\int_{0}^{\pi }{{{e}^{{{\sin }^{2}}(\pi -x)}}{{\cos }^{3}}(\pi -x)\,}dx\]                     ?..(i)            \[\Rightarrow I=-\int_{0}^{\pi }{{{e}^{{{\sin }^{2}}x}}{{\cos }^{3}}x\,dx}\]                  ?..(ii)                 Adding (i) and (ii), we get \[2I=0\] Þ \[I=0\].


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