Example 2.40. $I=\int_{0}^{0.5} \frac{d x}{x \ln ^{2} x}$.
Solution.

$$
\begin{aligned}
& I=\lim _{\varepsilon \rightarrow+0} \int_{\varepsilon}^{0.5} \frac{d(\ln x)}{\ln ^{2} x}=\left.\lim _{\varepsilon \rightarrow+0} \frac{-1}{\ln x}\right|_{\varepsilon} ^{0.5}= \\
& =-\frac{1}{\ln 0.5}+\lim _{\varepsilon \rightarrow+0} \frac{1}{\ln \varepsilon}=\frac{1}{\ln 2} \approx 1.443
\end{aligned}
$$