Question

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

Answer 1

$\begin{array}{c}\int \frac{2x+3}{{\left(x-1\right)}^2}dx\\ \int \frac{2x+2+1}{\left(x-1\right)\left(x+1\right)}dx\\ \Rightarrow \int \frac{2\left(x+1\right)}{\left(x-1\right)\left(x+1\right)}dx+\int \frac{1}{{\left(x-1\right)}^2}dx\\ \Rightarrow \int \frac{2}{x-1}dx+\int \frac{1}{{\left(x-1\right)}^2}dx\\ \Rightarrow \int 2\log \left(x-1\right)-\frac{1}{x-1}+c\end{array}$