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JAMIA MILLIA ISLAMIA Jamia Millia Islamia Solved Paper-2008

  • question_answer
        If\[f(x)\]is defined\[[-2,2]\]by\[f(x)=4{{x}^{2}}-3x+1\] and\[g(x)=\frac{f(-x)-f(x)}{{{x}^{2}}+3},\]then\[\int_{-2}^{2}{g(x)}dx\]is equal to

    A)  64                                         

    B)  \[-48\]

    C)  0                                            

    D)  24

    Correct Answer: C

    Solution :

                    Given that, \[f(x)=4{{x}^{2}}-3x+1,g(x)=\frac{f(-x)-f(x)}{{{x}^{2}}+3}\] \[\therefore \] \[g(x)=\frac{(4{{x}^{2}}+3x+1)-(4{{x}^{2}}-3x+1)}{{{x}^{2}}+3}\]                 \[=\frac{6x}{{{x}^{2}}+3}\] Now,     \[g(-x)=-\frac{6x}{{{x}^{2}}+3}\]                 \[=-g(x)\] Which is an odd function \[\therefore \]  \[\int_{-2}^{2}{g(x)}\,dx=0\]


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