TITLE: How to explain the behaviour of $\int x^n dx$ near $n = -1$ to a layman?
QUESTION [8 upvotes]: From both sides, this approaches infinity, but when evaluated exactly at $n = -1$, yields $\ln (x)$.
This seems similar to the behaviour of solutions to linear ODEs with characteristic polynomials (as the determinant approaches 0, a factor of $x$ appears next to the repeated root).
How can I explain this phenomenon to a layman? Obviously, I could show a proof of both, but that is probably not enough to be satisfactory. How does an infinity turn into a logarithm at the drop of a hat?
As an aside, is there a name for this behaviour?

REPLY [3 votes]: Another thought.  If $0<a<b$, $n\ne -1$, then
$$
\int_a^b x^n \;dx = \frac{b^{n+1}-a^{n+1}}{n+1}
$$
and the limit of that as $n \to -1$ is $\ln b - 
\ln a$.  Not infinity.
So, in a certain sense, your confusion comes from ignoring the constant of integration.