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

∴ $\frac{x}{(x+1)(x+2)}=\frac{-1}{(x+1)}+\frac{2}{(x+2)}$

$\Rightarrow \int \frac{x}{(x+1)(x+2)}dx = \int \frac{-1}{(x+1)}+\frac{2}{(x+2)}dx$

= $- \log \mid x+1 \mid +2\log \mid x+2 \mid+C$

= $\log(x+2)^2 -\log \mid x+1 \mid+C$

= $\log(\frac{x+2)^2}{ (x+1)}+C$