I am doing a Reduction of Order problem that requires an Integrating Factor. I understand how to do it [almost] perfectly fine. I do not understand a couple steps in the integration processes.
$$xy''+y'=0; y_1=ln(x)$$ The answer is $y_2=1$. Keep in mind this ISN'T an IVP nor is any interval of validity implied or given.
I get to the integrating factor computed by...
$$μ(x)=e^{\int\frac{(2+ln(x))dx}{xln(x)}}$$ $$=e^{ln(ln^2(x))}・e^{ln|x|}$$ $$=|x|ln^2(x)$$
This $μ(x)$ causes trouble with one of the final integrals... $$\int du=C_1\int\frac{dx}{|x|ln^2(x)}$$
This leads me to my question. Why do so many sources (such as Wolfram Alpha in the case of my $μ(x)$) drop the abs(...) from the end result of the integral? I distinctly recall being taught in Calculus classes to keep the abs(...) when it applies.
Linked here is my overall work until the point where I am stuck.
hint when you write $\ln^2 (x) $, it means that $x>0$ so $|x|=x $.