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

  • question_answer
    Suppose f is such that \[f(-x)=-f(x)\] for every real x and \[\int_{\,0}^{\,1}{f(x)\,dx=5,}\] then \[\int_{\,-\,1}^{\,0}{f(t)\,dt=}\]                     [MP PET 2000]

    A)                 10          

    B)                 5

    C)                 0             

    D)                 ? 5

    Correct Answer: D

    Solution :

               Given \[f(-x)=-f(x)\]            We know that, \[\int_{-a}^{a}{f(x)dx=0=\int_{-a}^{0}{f(x)dx+}\int_{0}^{a}{f(x)dx}}\]                    \[\Rightarrow \int_{-1}^{0}{f(x)\,dx+}\int_{0}^{1}{f(x)\,dx=0}\Rightarrow \int_{-1}^{0}{f(x)dx=-5}\]                                 \[\Rightarrow \int_{-1}^{0}{f(t)\,dt=-5}\].


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