Question

Integrate the rational function $\frac{2x}{x^2+3x+2}$

Answer 1

Let $\frac{2x}{x^2+3x+2}=\frac{A}{(x+1)}+\frac{B}{(x+2)}$
$\Rightarrow 2x=A(x+2)+B(x+1)$ ............. (1)
Substituting $x=1$ and 2 in equation (1), we obtain
$A=2$ and $B=4$
$\therefore \frac{2x}{(x+1)(x+2)}=\frac{-2}{(x+1)}+\frac{4}{(x+2)}$
$\Rightarrow \int \frac{2x}{(x+1)(x+2)}dx=\int \{\frac{4}{(x+2)}-\frac{2}{(x+1)}\}dx$
$=4\log |x+2|-2\log |x+1|+C$