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JEE Main & Advanced Mathematics Indefinite Integrals Question Bank Self Evaluation Test - Integrals

  • question_answer
    \[\int_{0}^{1}{[f(x)g''(x)-f''(x)g(x)]dx}\] is equal to: [Given f(0) = g (0) = 0]

    A) \[f(1)g(1)-f(1)g'(1)\]

    B) \[f(1)g'(1)+f'(1)g(1)\]

    C) \[f(1)g'(1)-f'(1)g(1)\]

    D) None of these  

    Correct Answer: C

    Solution :

    [c] Integrating by parts. \[\int{f(x)g''(x)dx-\int{f''(x)g(x)dx}}\] \[=f(x)g'(x)-\int{f'(x)g'(x)dx}\] \[-f'(x)g(x)+\int{f'(x)g'(x)dx}\]             \[=f(x)g'(x)-f'(x)g(x)\]             Hence, \[\int_{0}^{1}{f(x)g''(x)dx-\int_{0}^{1}{f''(x)g(x)dx}}\]             \[=f(1)g'(1)-f'(1)g(1)-f(0)g'(0)+f'(0)g(0)\]             \[=f(1)g'(1)-f'(1)g(1)\]


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