∫exx+2x+42dx=∫exx2+4x+4(x+4)2dx=∫exx(x+4)(x+4)2+4(x+4)2dx=∫exxx+4+4(x+4)2dx=exxx+4+C

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$\int e^{x} {(\frac{x + 2}{x + 4})}^{2} dx =$

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$e^{x} (\frac{x}{x + 1}) + c$

$e^{x} (\frac{x}{x + 4}) + c$

$e^{x} (\frac{x}{x + 3}) + c$

$e^{x} (\frac{x}{x + 2}) + c$

answer is D.

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Detailed Solution

$\int e^{x} {(\frac{x + 2}{x + 4})}^{2} d x$
$= \int e^{x} [\frac{x^{2} + 4 x + 4}{(x + 4)^{2}}] d x$
$\begin{array}{l} = \int e^{x} [\frac{x (x + 4)}{(x + 4)^{2}} + \frac{4}{(x + 4)^{2}}] d x \\ = \int e^{x} [\frac{x}{x + 4} + \frac{4}{(x + 4)^{2}}] d x = e^{x} (\frac{x}{x + 4}) + C \end{array}$