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?


Simple: $\quad -\log x=_0 o\Bigl(\dfrac1{\sqrt x}\Bigr)$ and the integral of $\dfrac 1{\sqrt x}$ on [0,1] is convergent.
