It was hard to come up with a good name for this post. Feel free to change it.
I am trying to solve the following problem, given to me by a fellow student, as we study for upcoming qualifying exams. No idea where he got it from:
"Determine the values of $p$ and $q$ for which the integral
$$\int_0^\infty \dfrac{x^{p-1}}{1+x^q}dx$$
exists as a Lebesgue integral. Hint: there should be a range of values."
Of course, "exists as a Lebesgue integral" really means "is finite".
I have observed the following: When $p-1<0$, we have a problem as $x\to0$, similarly when $q<0$.
But I'm quite sure that simple inspection is not sufficient, and I have no idea how to prove that all other values work. I in fact doubt that I have found everything, and think that perhaps those ranges may be fine, since things like $\int_0^1\frac{x}{\ln x}dx$ exist as Lebesgue integrals, even though problems arise as $x\to 0$ at first glance.
Usually, when I prove a Lebesgue integral exists, I resort to sums, the integral test, monotone convergence theorem, etc., but I feel like that doesn't lead anywhere here, since the integrand may or may not be L-integrable.
This is precisely the type of problem I struggle with regularly. Also, be forewarned: I ask a lot of questions, usually. I will be back tomorrow morning to check in and see what I can learn. Many thanks for any and all help, y'all.