Ok, here is the function at hand:
$ f(x)=\frac{1-e^{-x}}{x}-\frac{1}{1+x} $
My problem is to determine if the Lebesgue integral on the interval $(0,\infty)$ exists, and if it does, whether or not it is finite.
My first issue is some confusion on what it means for the integral to exist in the first place. I feel like finiteness implies existence and vice versa, but the question would not be posed in such a manner if it did not allow for infinite integrals.
In doing these types of problems, I am less interested in the tools to explicitly calculate the integral, but more in understanding the theory. So, if someone could help a self-studier like myself with breaking this problem down, I would be forever grateful.
Specifically, any detail as to the specific justification of why the integral exists (sequences of simple functions, continuous function on a finite interval, etc.) is especially helpful.
From here I am going to attempt a number of similar exercises, but it would go a long way to see how one such exercise can be done.