Example 2.39. $I=\int_{0}^{1} \frac{d x}{x^{2}-5 x+4}$.

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Example 2.39. $I=\int_{0}^{1} \frac{d x}{x^{2}-5 x+4}$.
Solution. Decompose the integrand into a sum of the simplest rational fractions of the first type, and then use formula (2.17)

$$
\begin{gathered}
I=\int_{0}^{1} \frac{d x}{(x-1)(x-4)}=\frac{1}{3} \int_{0}^{1}\left(\frac{1}{x-4}-\frac{1}{x-1}\right) d x= \\
=\lim _{\varepsilon \rightarrow+0} \frac{1}{3} \int_{0}^{1-\varepsilon}\left(\frac{1}{x-4}-\frac{1}{x-1}\right) d x= \\
=\left.\frac{1}{3} \lim _{\varepsilon \rightarrow+0} \ln \left|\frac{x-4}{x-1}\right|\right|_{0} ^{1-\varepsilon}=\frac{1}{3}\left(\lim _{\varepsilon \rightarrow+0} \ln \left|\frac{-3-\varepsilon}{-\varepsilon}\right|-\ln 4\right)=\infty
\end{gathered}
$$

the integral diverges.