Question

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

Answer 1

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

$=\int \frac{1}{x-2}dx-\int \frac{1}{x-1}dx$

$=ln|x-2|-ln|x-1|+c$

$=ln\left|\frac{x-2}{x-1}\right|+c$