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. $\frac{e^{x}}{x+3}+C$
C. $\frac{1}{(x+4)^{2}}+C$
D. $\frac{\mathrm{e}^{\mathrm{x}}}{(\mathrm{x}+4)^{2}}+\mathrm{C}$
Solution:
$\int \frac{x+3}{(x+4)^{2}} e^{x} d x$
$=\int \frac{x+4}{(x+4)^{2}} e^{x} d x-\int \frac{1}{(x+4)^{2}} e^{x} d x$
$=\int e^{x}\left(\frac{1}{x+4} d x-\frac{1}{(x+4)^{2}} d x\right)$
$\left[\because f(x)=\frac{1}{x+4} ; f^{\prime}(x)=-\frac{1}{(x+4)^{2}}\right]$
$=e^{x}\left(\frac{1}{x+4}\right)+c$
$\left.\because\left\{\int e^{x} f(x)+f^{\prime} x\right]=e^{x} f(x)\right\}$