Question

Integrate the rational function $\frac{x}{(x-1)(x-2)(x-3)}$

Answer 1

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