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

  • question_answer
    \[\int{{{\left( \frac{x+2}{x+4} \right)}^{2}}{{e}^{x}}\,\,dx}\] is equal to   [AMU 2000]

    A)                 \[{{e}^{x}}\left( \frac{x}{x+4} \right)+c\]

    B)                 \[{{e}^{x}}\left( \frac{x+2}{x+4} \right)+c\]

    C)                 \[{{e}^{x}}\left( \frac{x-2}{x+4} \right)+c\]

    D)   \[\left( \frac{2x{{e}^{x}}}{x+4} \right)+c\]

    Correct Answer: A

    Solution :

     \[I=\int{{{\left( \frac{x+2}{x+4} \right)}^{2}}{{e}^{x}}dx}\]\[=\int{{{e}^{x}}\left[ \frac{{{x}^{2}}+4x+4}{{{(x+4)}^{2}}} \right]}\,dx\]                 \[\Rightarrow I=\int{{{e}^{x}}\left[ \frac{x(x+4)}{{{(x+4)}^{2}}}+\frac{4}{{{(x+4)}^{2}}} \right]\,dx}\]               \[={{e}^{x}}\left[ \frac{x}{x+4}+\frac{4}{{{(x+4)}^{2}}} \right]\,dx\]\[={{e}^{x}}\left( \frac{x}{x+4} \right)+c\].


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