I want to show that the solution of a BVP: $\ u(x) = \ln{\frac{1+x}{1-x}}$ is in $L^2(0,1)$, so I need to show that the integral $$\int\limits_0^1\ln^2|\frac{x+1}{x-1}|dx < \infty$$
Just looking at the function, however, it's not even defined in the interval $[0,1]$, right? So can this not actually be a solution to the BVP? Even more generally about the integral itself, does that make it just $0$? And if not, can someone explain how an integral of a function can be defined in a region when the function itself is not?
Also, I know that there exist integral calculators online and have looked this up, specifically using: https://www.integral-calculator.com/
If you put in the given integral it says the integral could not be found and then gives a complex number approximation. What should be made of this? I have taken a complex variables course but don't really see how you could solve this using Residue Calculus since we don't have any symmetries.
EDIT: Note I had originally forgot the absolute value signs - my mistake.