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 :
\[I=\int{{{\left( \frac{x+2}{x+4} \right)}^{2}}{{e}^{x}}dx}\]\[=\int{{{e}^{x}}\left[ \frac{{{x}^{2}}+4x+4}{{{(x+4)}^{2}}} \right]}\,dx\] \[\Rightarrow I=\int{{{e}^{x}}\left[ \frac{x(x+4)}{{{(x+4)}^{2}}}+\frac{4}{{{(x+4)}^{2}}} \right]\,dx}\] \[={{e}^{x}}\left[ \frac{x}{x+4}+\frac{4}{{{(x+4)}^{2}}} \right]\,dx\]\[={{e}^{x}}\left( \frac{x}{x+4} \right)+c\].You need to login to perform this action.
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