Question

Integrate the rational function $\frac{5x}{(x+1)(x^2-4)}$

Answer 1

$\frac{5x}{(x+1)(x^2-4)}=\frac{5x}{(x+1)(x+2)(x-2)}$
Let $\frac{5x}{(x+1)(x+2)(x-2)}=\frac{A}{(x+1)}+\frac{B}{(x+2)}+\frac{C}{(x-2)}$
$\Rightarrow 5x=A(x+2)(x-2)+B(x+1)(x-2)+C(x+1)(x+2)$ ......... (1)
Substituting $x=1,2,$ and $-2$ respectively in equation (1), we obtain
$A=\frac{5}{3},B=-\frac{5}{2}$, and $C=\frac{5}{6}$
$\therefore \frac{5x}{(x+1)(x+2)(x-2)}=\frac{5}{3(x+1)}-\frac{5}{2(x+2)}+\frac{5}{6(x-2)}$
$\Rightarrow \int \frac{5x}{(x+1)(x^2-4)}dx=\frac{5}{3}\int \frac{1}{(x+1)}dx-\frac{5}{2}\int \frac{1}{(x+2)}dx+\frac{5}{6}\int \frac{1}{(x-2)}dx$
$=\frac{5}{3}\log |x+1|-\frac{5}{2}\log |x+2|+\frac{5}{6}\log |x-2|+C$