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?


The key is how fast the function is diverging.
Regarding your two examples, $-\log$ is going really fast close to the $y$-axis so it is integrable, but not $x\mapsto \frac {1}{x}$.


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*You have $$\int_a^1 -\log(x)\mathrm dx=a(1-\log(a))+1\xrightarrow[a\to 0^+]{} 1<\infty.$$ So this function is integrable.

*You have $$\int_a^1 \frac 1x\mathrm dx=-1+\frac 1{a^2}\xrightarrow[a\to 0^+]{} +\infty.$$ So this function is not integrable.
Regarding your second question, $\log$ functions are absolutely not the only one. To convince yourself, take for instance $x\mapsto \frac 1{\sqrt{x}}$ on $(0,1)$.
