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

  • question_answer
    If \[\int_{-1}^{4}{f(x)\,dx}=4\] and \[\int_{2}^{4}{(3-f(x))\,dx=7,}\] then the value of \[\int_{2}^{-1}{f(x)\,dx}\]  is

    A)                 2             

    B)                 ? 3

    C)                 ? 5         

    D)                 None of these

    Correct Answer: C

    Solution :

               We have \[\int_{2}^{4}{(3-f(x))dx=7\Rightarrow 6-\int_{2}^{4}{f(x)dx=7}}\]                    Þ \[\int_{2}^{4}{f(x)dx=-1}\].                    Now, \[\int_{2}^{-1}{f(x)dx=-\int_{-1}^{2}{f(x)dx=-\left[ \int_{-1}^{4}{f(x)dx+\int_{4}^{2}{f(x)dx}} \right]}}\]                                                \[=-\left[ \int_{-1}^{4}{f(x)dx-\int_{2}^{4}{f(x)dx}} \right]=-(4+1)=-5\].


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