Question

$\int \frac{3x-1}{(x-1)(x-2)(x-3)}dx$

Answer 1

Let

$\frac{3x-1}{(x-1)(x-2)(x-3)}=\frac{A}{(x-1)}+\frac{B}{(x-2)}+\frac{C}{(x-3)}$

$(3x-1)=A(x-2)(x-3)+B(x-1)(x-3)+C(x-1)(x-2)$

Put $x=1,A=1$

$x=2,B=-5$

$x=3,C=4$

$\int \frac{x-1}{(x-1)(x-2)(x-3)}dx=\int (\frac{1}{x-1}-\frac{5}{x-2}+\frac{4}{x-3})dx$

$=\int \frac{1}{x-1}dx-5\int \frac{1}{x-2}dx+4\int \frac{1}{x-3}dx$

$=\log |x-1|-5\log |x-2|+4\log |x-3|+C$