What is the content you want to add? What problem is it the solution to?
For a fully rigorous solution, we need to add a proof that
$$\int_0^\infty dx \frac{\ln x}{1 + x^2} = 0,$$
which is indeed true according to WolframAlpha.
This could be done by integrating $\frac{{\left( \ln z \right)}^2}{1 + z^2}$ along the same contour.
Additional context
Perhaps we could generalize this as $$I_n := \int_0^\infty dx \frac{{\left( \ln x \right)}^n}{1 + x^2},$$ for which we may be able to find a recurrence relation for by integrating $$\frac{{\left( \ln z \right)}^{n + 1}}{1 + z^2}.$$
What is the content you want to add? What problem is it the solution to? For a fully rigorous solution, we need to add a proof that $$\int_0^\infty dx \frac{\ln x}{1 + x^2} = 0,$$ which is indeed true according to WolframAlpha. This could be done by integrating $\frac{{\left( \ln z \right)}^2}{1 + z^2}$ along the same contour.
Additional context Perhaps we could generalize this as $$I_n := \int_0^\infty dx \frac{{\left( \ln x \right)}^n}{1 + x^2},$$ for which we may be able to find a recurrence relation for by integrating $$\frac{{\left( \ln z \right)}^{n + 1}}{1 + z^2}.$$