Why is $-\log(x)$ integrable over the interval $[0, 1]$ but $\frac{1}{x}$ not integrable? I don't understand why some functions that contain a singularity in the domain of integration are integrable but others are not.
For example, consider $f(x) = -\log(x)$ and $g(x) = \frac{1}{x}$ on the interval $[0, 1]$. These functions look very similar when they are plotted but only $f(x)$ can be integrated.


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*What is the precise mathematical reason(s) that makes some functions with singularities integrable while others are not?

*Are $\log$ functions the only functions with singularities that can be integrated or are there other types of functions with singularities that can be integrated?


Think about it this way - what's the inverse?
$$y = \frac{1}{x}; x = \frac{1}{y}$$
$$y = -\log x; x = e^{-y}$$
Looking at it this way, it's clear that as $y$ shoots off to infinity, $x$ approaches zero much faster in one case than in the other.
