How do I know if a improper integral only sometimes converges?

Question

Given that

$0<f(x)<\displaystyle\frac{1}{x}$ for $0<x<1$,

how do I know that $\int _0^1f(x)dx$ only diverges for some functions, and not for others? Is there a counter-example out there where this integral diverges, and if so is there a systematic way of finding such counter-examples?

Answer 1

diverges for $f(x) = \frac{1}{2x}.$ Converges for $f(x) = \frac{1}{2}.$

Answer 2

Sure. Think about extreme examples. In this case, the extreme examples are $\frac{1}{x}$ and $0$ (these aren't actually allowed, since the inequality is strict, but we'll get to that later). $\int_0^1\frac{1}{x}dx$ diverges. $\int_0^10dx$ converges, because it isn't actually an improper integral. That gives you a clue for what to look for - for something that diverges, we want something "like" $\frac{1}{x}$, and for something that converges we want something that isn't improper.

First: Something "like" $\frac{1}{x}$ that's a little smaller - how about half of it? So, $\frac{1}{2x}$? You can check for yourself that $\int_0^1\frac{1}{2x}dx$ diverges.

Second: Something "like" $0$ that's a little bigger - how about $0.1$? $\int_0^10.1dx$ isn't improper, so it definitely converges.