Integration $\displaystyle\int \frac{x}{x^2-5x+6}dx$

Question

Evaluate the Integral:

$$\int \frac{x}{x^2-5x+6}dx$$

I solved twice and once I got $$3\log\left|x-3\right|-2\log\left|x-2\right|+C$$ and I tried again and changed one step and I got $$-2\log\left|x-3\right|-3\log\left|x-2\right|+C$$

which one is correct? and why ?

Answer 1

Convert Integrand to Partial fractions

$$\frac{x}{x^2-5x+6}=\frac{3}{x-3}-\frac{2}{x-2}$$

$$\begin{align}I&=\int \frac{x}{x^2-5x+6}dx\\ &=\int\frac{3}{x-3}dx-\int\frac{2}{x-2}dx\\ &=3\ln|x-3|-2\ln|x-2|+C\\ \end{align}$$

Try again and I'm sure you'll get this!