Question

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

Answer 1

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