Mark the correct alternative in each of the following:
Question:

Mark the correct alternative in each of the following:

Evaluate $\int \frac{x+3}{(x+4)^{2}} e^{x} d x=$

A. $\frac{e^{x}}{x+4}+C$

B. $\frac{\mathrm{e}^{\mathrm{x}}}{\mathrm{x}+3}+\mathrm{C}$

C. $\frac{1}{(x+4)^{2}}+C$

D. $\frac{e^{x}}{(x+4)^{2}}+C$

Solution:

$\int \frac{x+3}{(x+4)^{2}} e^{x} d x$

$=\int \frac{x+4}{(x+4)^{2}} e^{x} d x-\int \frac{1}{(x+4)^{2}} e^{x} d x$

$=\int e^{x}\left(\frac{1}{x+4} d x-\frac{1}{(x+4)^{2}} d x\right)$

$\left[\because f(x)=\frac{1}{x+4} ; f^{\prime}(x)=-\frac{1}{(x+4)^{2}}\right]$

$=e^{x}\left(\frac{1}{x+4}\right)+c$

$\left.\because\left\{\int e^{x} f(x)+f^{\prime} x\right]=e^{x} f(x)\right\}$

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