I was playing around with some integrals, and this question popped into my head:
What functions exist such that the following is true? $$\int f(g(x))\;\mathrm dx = f\left(\int g(x)\;\mathrm dx\right)$$
There there's the obvious example of $f(x) = x, \;g(x)=e^x$, but I was wondering if others exist.
EDIT 1: As pointed out in the comments, this is true for any $g$ if $f(x) = x$. But, this is sort of trivial--I'd really like to know about for other assignments of $f$...
My question is twofold:
- Are there known functions that satisfy this equality?
- What sort of topic in math would this fall under? (e.g. Abstract Algebra, Differential/Integral equations, etc.)
I'd also accept an answer to a similar, but slightly different question, as phrased in the comments by user1551; if it's easier/more feasible to answer:
Find a pair of functions $f$ and $g$ such that $\int_a^b f(g(x))dx=f\left(\int_a^bg(x)dx\right)$ for any interval $[a,b]$