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JEE Main & Advanced Mathematics Indefinite Integrals Question Bank Self Evaluation Test - Integrals

  • question_answer
    Let \[f:R\to R\] and \[g:R\to R\] be continuous functions. Then the value of \[\int\limits_{-\frac{\pi }{2}}^{\frac{\pi }{2}}{\{f(x)+f(-x)\}\{g(x)-g(-x)\}dx}\] is

    A) \[f(x)g(x)\]

    B) \[f(x)+g(x)\]

    C) 0

    D) None of theses

    Correct Answer: C

    Solution :

    [c] Let \[\phi (x)=\left\{ f(x)+f(-x) \right\}\left\{ g(x)-g(-x) \right\}\] \[\phi \,(-x)=\left\{ f(-x)+f(x) \right\}\left\{ g(-x)-g(x) \right\}\] \[=-\left\{ f(x)+f(-x) \right\}\left\{ g(x)-g(-x) \right\}=-\phi (x)\] \[\therefore \,\,\,\,\varphi (x)\] is an odd function \[\Rightarrow \int\limits_{\frac{-\pi }{2}}^{\frac{\pi }{2}}{\phi (x)dx=0}\]


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