Answer
$$5\ln |x-3|-3\ln |x-2|+C$$
Work Step by Step
Given $$\int \frac{(2 x-1) d x}{x^{2}-5 x+6}$$
Since
\begin{align*}
\frac{(2 x-1) }{x^{2}-5 x+6}&= \frac{(2 x-1) }{(x-2)(x-3)}\\
&=\frac{A}{x-2}+\frac{B}{x-3}\\
&=\frac{A(x-3)+B(x-2)}{(x-2)(x-3)}\\
2x-1&=A(x-3)+B(x-2)
\end{align*}
Then
\begin{align*}
\text{at } x&=3\ \ \ \ \ A=-3 \\
\text{at } x&=2\ \ \ \ \ B=5
\end{align*}
Hence
\begin{align*}
\int \frac{(2 x-1) d x}{x^{2}-5 x+6}&=\int \frac{-3}{x-2}dx+\int \frac{5}{x-3}dx\\
&=5\ln |x-3|-3\ln |x-2|+C
\end{align*}
