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

The correct option is C $\frac{e^x}{(x+4)}+C$

$\int \frac{(x+3)e^x}{(x+4{)}^2}dx=\int \frac{(x+4)e^x-e^x}{(x+4{)}^2}dx=\int \frac{e^x}{(x+4)}dx-\int \frac{e^x}{(x+4{)}^2}dx\text{By applying integration by parts in the first part}=\frac{e^x}{(x+4)}+\int \frac{e^x}{(x+4{)}^2}dx-\int \frac{e^x}{(x+4{)}^2}dx=\frac{e^x}{(x+4)}+C$