Question

$\int_{}^{}\frac{(x + 3)e^{x}}{(x + 4)^{2}}dx$ is equal to

• A. $\frac{e^{x}}{x + 4} + c$
• B. $\frac{e^{x}}{x + 3} + c$
• C. $\frac{1}{(x + 4)^{2}} + c$
• D. $\frac{e^{x}}{(x + 4)^{2}} + c$

Answer 1

Answer: (A)

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

Answer 2

$\int_{}^{}\frac{(x + 3)e^{x}}{(x + 4)^{2}}dx = \int_{}^{}\frac{(x + 4 - 1)e^{x}}{(x + 4)^{2}}dx = \int_{}^{}\frac{(x + 4)e^{x}}{(x + 4)^{2}}dx - \int_{}^{}{\frac{e^{x}}{(x + 4)^{2}}dx}$ $= \int_{}^{}{\frac{e^{x}}{(x + 4)}dx - \int_{}^{}{\frac{e^{x}}{(x + 4)^{2}}dx}}$

$= \left\lbrack \frac{e^{x}}{(x + 4)^{}} + \int_{}^{}{\frac{e^{x}}{(x + 4)^{2}}dx} \right\rbrack - \int_{}^{}{\frac{e^{x}}{(x + 4)^{2}}dx + c}$ $= \frac{e^{x}}{x + 4} + c$