Question

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

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

Answer 1

Answer: (C)

Let $I=\int \frac{x+3}{(x+4{)}^2}{e}^{x}dx$
$=\int e^x\left[\frac{x+4-1}{(x+4{)}^2}\right]dx$
$=\int e^x\left[\frac{1}{x+4}+\frac{-1}{(x+4{)}^2}\right]dx$
$=\int e^x\frac{1}{x+4}dx+\int e^x\left(\frac{-1}{(x+4{)}^2}\right)dx$
$=\frac{1}{x+4}{e}^x-\int e^x\cdot \left(\frac{-1}{(x+4{)}^2}\right)dx+$
$\int e^x\left(-\frac{1}{(x+4{)}^2}\right)dx$
$=\frac{e^x}{x+4}+C$