Question

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

Answer 1

$$\begin{gathered} \int \left[\frac{2x-1}{{\left(x-1\right)}^2}\right]dx \\ =\int \left[\frac{2x-2+2-1}{{\left(x-1\right)}^2}\right]dx \\ =\int \left(\frac{2\left(x-1\right)}{{\left(x-1\right)}^2}+\frac{1}{{\left(x-1\right)}^2}\right)dx \\ =2\int \frac{dx}{x-1}+\int {\left(x-1\right)}^{-2}dx \\ =2\ln \left|x-1\right|+\frac{{\left(x-1\right)}^{-2+1}}{-2+1}+\mathrm{C} \\ =2\ln \left|x-1\right|-\frac{1}{x-1}+\mathrm{C} \end{gathered}$$