Question

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

Answer 1

$\int \frac{x}{(x+1)(x+2)}.dx$
Using partial fraction
$\Rightarrow \frac{x}{(x+1)(x+2)}=\frac{A}{(x+1)}+\frac{B}{(x+2)}$
$\Rightarrow x=A(x+2)+B(x+1)$
$\Rightarrow -1=A(-1+2)+B(-1+1)$
$\Rightarrow A=-1$
$\Rightarrow -2=A(-2+2)+B(-2+1)$
$\Rightarrow -2=-B\Rightarrow B=2$
$\Rightarrow \frac{x}{(x+1)(x+2)}=\frac{-1}{(x+1)}+\frac{2}{(x+2)}$
Now,
$\int \frac{x}{(x+1)(x+2)}.dx$
$=\int \frac{-1}{(x+1)}.dx+\int \frac{2}{(x+2)}.dx$
$=-\log |x+1|+2\log |x+2|+C$
$=-\log |x+1|+\log |x+2{|}^2+C$
$=\log \left|\frac{(x+2{)}^2}{x+1}\right|+C$
Where C is constant of integration