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

  • question_answer
    If \[f:R\to R\] and \[g:R\to R\] are one to one, real valued functions, then the value of the integral \[\int_{\,-\pi }^{\,\pi }{[f(x)+f(-x)]\,[g(x)-g(-x)]\,dx}\] is                 [DCE 2001; MP PET 2004]

    A)                 0             

    B)                 \[\frac{8}{3}\]

    C)                 1             

    D)                 None of these

    Correct Answer: A

    Solution :

               Let \[\varphi (x)=[f(x)+f(-x)][g(x)-g(-x)]\]            then, \[\varphi (-x)=[f(-x)+f(x)]\,[g(-x)-g(x)]\]                 \[\therefore \int_{-\pi }^{\pi }{\varphi (x)dx=0}\]Þ\[\int_{-\pi }^{\pi }{[f(x)+f(-x)][g(x)-g(-x)]dx=0}\].


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