Obliczamy całkę metodą podstawiania: \[ \begin{split} \int \frac{x-5}{x^2-10x+9}dx &=\int \frac{(x-5)dx}{x^2-10x+9}=\\[6pt] &=\begin{vmatrix} t=x^2-10x+9 \\ dt=(2x-10)dx \\ dt=2(x-5)dx \\ \frac{dt}{2}=(x-5)dx\end{vmatrix}=\\[6pt] &=\int \frac{\frac{dt}{2}}{t}=\\[6pt] &=\int \frac{dt}{2t}=\\[6pt] &=\frac{1}{2}\int \frac{1}{t}dt=\\[6pt] &=\frac{1}{2}\ln |t|+C=\\[6pt] &=\frac{1}{2}\ln |x^2-10x+9|+C \end{split} \]