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CET Karnataka Engineering CET - Karnataka Engineering Solved Paper-2008

  • question_answer
    If \[\int{f(x)\,dx=g(x),}\] then \[\int{f(x)\,g=g(x)\,dx}\] is equal to

    A)  \[\frac{1}{2}{{f}^{2}}(x)\]                           

    B)  \[\frac{1}{2}{{g}^{2}}(x)\]

    C)  \[\frac{1}{2}{{[g'(x)]}^{2}}\]                      

    D)  \[f'(x)g(x)\]           

    Correct Answer: B

    Solution :

    Given,  \[\int{f(x)}dx=g(x)\] \[\therefore \]  \[\int{f\underset{II}{\mathop{(x)}}\,}g\underset{I}{\mathop{(x)}}\,dx=g(x)\int{f(x)\,dx}\]                                 \[-\int{[g'(x)\int{f(x)dx]dx}}\]                 \[=g(x)g(x)-\int{g'(x)g(x)dx}\]                 \[={{[g(x)]}^{2}}-\frac{[g{{(x)}^{2}}]}{2}\]                 \[=\frac{{{g}^{2}}(x)}{2}\]


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