Question

Integrate the rational function $\frac{3x-1}{(x+2{)}^2}$

Answer 1

Let $\frac{3x-1}{(x+2{)}^2}=\frac{A}{(x+2)}+\frac{B}{(x+2{)}^2}$
$\Rightarrow 3x-1=A(x+2)+B$
Equating the coefficient of $x$ and constant term, we obtain
$A=3,2A+B=-1\Rightarrow B=-7$
$\therefore \frac{3x-1}{(x+2{)}^2}=\frac{3}{(x+2)}-\frac{7}{(x+2{)}^2}$
$\Rightarrow \int \frac{3x-1}{(x+2{)}^2}dx=3\int \frac{1}{(x+2)}dx-7\int \frac{x}{(x+2{)}^2}dx$
$=3\log |x+2|-7\left(\frac{-1}{(x+2)}\right)+C$
$=3\log |x+2|+\frac{7}{(x+2)}+C$