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  5. Definite Integrals

12th Class Mathematics Definite Integrals Question Bank Critical Thinking

  • question_answer
    Let \[f:R\to R\] and \[g:R\to R\] be continuous functions, then the value of the integral \[\int_{-\pi /2}^{\pi /2}{[f(x)+f(-x)]\,\,[g(x)-g(-x)]\,dx=}\] [IIT 1990; DCE 2000; MP PET 2001]

    A) \[\pi \]                                    

    B) 1

    C) \[-1\]                                       

    D) 0

    Correct Answer: D

    Solution :

    • Let \[h(x)=\{f(x)+f(-x)\}\{g(x)-g(-x)\}\]                   
    • \[h\,(-x)=\{f(-x)+f(x)\}\{g(-x)-g(x)\}\]                             
    • \[=-\{f(-x)+f(x)\}\{g(x)-g(-x)\}=-h\,(x)\]                   
    • Therefore, \[\int_{-\pi /2}^{\pi /2}{\,h(x)dx=0}\].


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