Question

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

Answer 1

Let $\frac{x}{(x+1)(x+2)}=\frac{A}{(x+1)}+\frac{B}{(x+2)}$
$\Rightarrow x=A(x+2)+B(x+1)$
Equating the coefficients of $x$ and constant term, we obtain
$A+B=1,2A+B=0$
On solving, we obtain
$A=-1$ and $B=2$
$\therefore \frac{x}{(x+1)(x+2)}dx=\frac{-1}{(x+1)}+\frac{2}{(x+2)}$
$\Rightarrow \int \frac{x}{(x+1)(x+2)}dx=\int \frac{-1}{(x+1)}+\frac{2}{(x+2)}dx$
$=-\log |x+1|+2\log |x+2|+C$
$=\log (x+2{)}^2-\log |x+1|+C$
$=\log \frac{(x+2{)}^2}{(x+1)}+C$