Question

Integrate the rational functions.
$\int \frac{3x-1}{(x+2{)}^2}dx.$

Answer 1

Let $\frac{3x-1}{(x+2{)}^2}=\frac{A}{(x+2)}+\frac{B}{(x+2{)}^2}\Rightarrow 3x-1=A(x+2)+B$
On equating the coefficients of x and constant term on both sides, we get
A=3 and 2A+B =-1
$\Rightarrow 2\left(3\right)+B=-1\Rightarrow B=-7\therefore \frac{3x-1}{(x+2{)}^2}=\frac{3}{(x+2)}-\frac{7}{(x+2{)}^2}\therefore \int \frac{3x-1}{(x+2{)}^2}=3\int \frac{1}{(x+2)}dx-7\int \frac{1}{(x+2{)}^2}dx=3log|x+2|-7\left(\frac{-1}{x+2}\right)+C=3log|x+2|+\frac{7}{x+2}+C$