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RAJASTHAN ­ PET Rajasthan PET Solved Paper-2009

  • question_answer
    If\[f(x)\]and\[g(x),x\in R\]are continuous functions, then value of integral \[\int_{-\pi /2}^{\pi /2}{[\{f(x)+f(-x)\}\{g(x)-g(-x)\}]}\,dx\]is

    A)  \[\pi \]                

    B)  \[\pi /2\]

    C)  \[1\]                

    D)  0

    Correct Answer: D

    Solution :

     Let \[I=\int_{-\pi /2}^{\pi /2}{\{f(x)+f(-x)\}\{g(x)-g(-x)\}dx}\] Again,let \[h(x)=\{f(x)+f(-x)\}\{g(x)-g(-x)\}\] \[\Rightarrow \] \[h(-x)=\{f(-x)+f(x)\}\{g(-x)-g(x)\}\] \[\Rightarrow \] \[h(-x)=-h(x)\] Hence,\[h(x)\]is an odd function. \[\therefore \] \[I=0\]


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