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JEE Main & Advanced Mathematics Indefinite Integrals Question Bank Integration by Parts

  • question_answer
    \[\int_{{}}^{{}}{[f(x)g''(x)-f''(x)g(x)]}\ dx\] is equal to      [MP PET 2001]

    A)                 \[\frac{f(x)}{g'(x)}\]

    B)                 \[f'(x)g(x)-f(x)g'(x)\]

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

    D)                 \[f(x)g'(x)+f'(x)g(x)\]

    Correct Answer: C

    Solution :

                    \[\int_{{}}^{{}}{[f(x)\,{g}''(x)-{f}''(x)\,g(x)]\,dx}\]                 \[=\int_{{}}^{{}}{f(x)\,{g}''(x)\,dx}-\int_{{}}^{{}}{{f}''(x)\,g(x)\,dx}\]                 \[=\left( f(x)\,{g}'(x)-\int_{{}}^{{}}{{f}'(x){g}'(x)\,dx} \right)-\left( g(x)\,{f}'(x)-\int_{{}}^{{}}{{g}'(x)\,{f}'(x)\,dx} \right)\]                 \[=f(x)\,{g}'(x)-{f}'(x)\,g(x).\]


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