Question

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

Answer 1

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

$\frac{3x-1}{(x-1)(x-2)(x-3)}=\frac{a}{x-1}+\frac{b}{x-2}+\frac{c}{x-b}$

$a=\frac{3\left(1\right)-1}{(1-2)(1-3)}=\frac{2}{-1(-2)}=1$

$b=\frac{3\left(2\right)-1}{(2-1)(2-3)}=\frac{5}{-1}=-5$

$c=\frac{3\left(3\right)-1}{(3-1)(3-2)}=\frac{8}{2\times 1}=4$

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

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

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