\(=\int\dfrac{6}{\left(x-1\right)\left(2x-3\right)}dx=\int\dfrac{12}{\left(2x-2\right)\left(2x-3\right)}dx\)
\(=\int\left(\dfrac{12}{2x-3}-\dfrac{12}{2x-2}\right)dx=\int\left(\dfrac{12}{2x-3}-\dfrac{6}{x-1}\right)dx\)
\(=6ln\left|2x-3\right|-6ln\left|x-1\right|+C\)