Say I have f(x) and then I have Σ(f(x))dx (I'm using Σ in place of an integral since there's no character for an integral that fits on one line), is there some formula relating Σ(f(x))dx to f^-1(x)? Sort of like the opposite of the derivative rule for inverse functions?

The closest that comes to mind is the sort of thing that can be found here: https://en.wikipedia.org/wiki/Integral_of_inverse_functions and http://mathworld.wolfram.com/InverseFunctionIntegration.html

Edited July 16, 201510 yr by ajb

Well thanks for that then.

Well thanks for that then.

Is that close to what you need?

Not really but if its the only thing I guess I'll take it. I was thinking more in terms of logic that the "proof without words" theorem would yield something nice like "the inverse of the integral of f-1(x) - the integral of f(x)." But, if that that complicated theorem is the best it gets then that's the best it gets.

I don't know what result you are looking for. However, you can play about with expressions and see if anything closer to what you need comes up.

The result I'm looking for is what I tried to say before. To understand it, I will clear up what I mean and say "inverse of the inverse." If I took f(x), and then took the inverse of that function, I would get inverse of f(x) or f-1(x). Now, if I take the inverse of the inverse, or the inverse of f-1(x), I get f(x).

With that concept in mind, it seems intuitive that there would be a formula which shows that the inverse of the integral of f-1(x) is the integral of f(x).

I know that's not actually right, because when I test it with f(x)=x^2 and sqrt(x) the theorem doesn't work.

The closest that comes to mind is the sort of thing that can be found here: https://en.wikipedia.org/wiki/Integral_of_inverse_functions and http://mathworld.wolfram.com/InverseFunctionIntegration.html

This might be useful for reduction formulae (writing an integral $I_n$ in terms of $I_{n \pm k}$).