Question

Find $\int \frac{xdx}{(x+1)(x+2)}$.

Answer 1

$\int \frac{xdx}{(x+1)(x+2)}$
Using partial fractions,
$\frac{x}{(x+1)(x+2)}=\frac{A}{x+1}+\frac{B}{x+2}$
$x=A(x+2)+B(x+1)$ put $x=-2$
Put $x=-1\therefore -2=0+B(-2+1)-2=-B$
$\therefore -1=A(-1+2)\Rightarrow A=-1B=2$
$\therefore \int \frac{xdx}{(x+1)(x+2)}=\int \frac{-1dx}{x+1}+\int \frac{2dx}{x+2}=-\log |x+1|+\log |x+2|$
$=\log \left|\frac{x+2}{x+1}\right|+C$.