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

  • question_answer
    If an ant derivative of \[f(x)\] is \[{{e}^{x}}\] and that of \[g(x)\] is \[\cos x,\]then \[\int{f(x)\cos x\,dx}+\int{g(x){{e}^{x}}dx=}\] [Kerala (Engg.) 2005]

    A)                 f(x)g(x)+c

    B)                 f(x)+g(x)+c

    C)                 \[{{e}^{x}}\cos x+c\]

    D)                 f(x) ? g(x)+c

    E)                 \[{{e}^{x}}\cos x+f(x)g(x)+c\]

    Correct Answer: C

    Solution :

                    \[\int{f(x)\cos xdx+\int{g(x){{e}^{x}}dx}}\]                                          \[=\int{{{e}^{x}}\cos xdx+\int{(-\sin x){{e}^{x}}dx}}\]                                                      \[=\frac{{{e}^{x}}}{2}(\cos x+\sin x)-\frac{{{e}^{x}}}{2}(\sin x-\cos x)+c\]                                                      \[=\frac{{{e}^{x}}}{2}(2\cos x)+c\]\[={{e}^{x}}\cos x+c\].


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