This question may sound simplistic but is there a mathematical process which lies directly beyond integration integration, or more specifically beyond finding the antiderivative? And by that I mean loosely what is the next step? I do apologize if this statement sounds vague to higher minds. But I have certainly been racking my brain over this for awhile, and perhaps someone enlightened to where I am going with this can offer some assistance. Much appreciated.
As you can have higher-order derivatives so you can have higher order integrations. i.e. The second order integration is basically how you get the solution to ##y''=f(x)## i.e. ##y=3x^2## ... will have indefinite integrals: ##\int y\;dx = x^3+c## ##\iint y\; dx\; dx = \frac{1}{4}x^4 + cx + d## However, it is possible to get carried away with this sort of thing. The integration does a bit more than finding the anti-derivative ... i.e. finding areas.
Yessir. I see your point. But what happens if I do infinite iterations of integrating starting with. Simple polynomial? I doubt if the end result would converge to anything algebraically inexpressible. But is there some higher and more abstract way to express what is going here?
There is no reason that the result, for an arbitrary integrand, would converge to anything in particular. The short answer to your question is "no". There are always more abstract ways to express things though.
I believe what OP is asking is if there's an iterative "operation" for integration, much as there is an iterative operation for addition in multiplication, or an iterative operation for multiplication in exponentiation. I'm no mathematician myself, but I believe that no such operation has been conceived, as we have no use of it. That said, though, you might want to look into transforms, like the Fourier Transform. While not what you're looking for, it's a good "next step", in my opinion.