I might approach the question by looking at it algebraically. 1. Differentiation has a no-nonsense compositional rule (i.e., the chain rule), and integration has no equivalent. 2. If you look at what you're integrating as formal field extensions to, say, the complexes, then the integral itself often cannot be represented in that field. (Example: integrals of functions on R(x), like 1/x, require you to extend your field transcendentally with a logarithm.) So not only do you have the problem of computing the integral, you need to compute where the integral even lives. Differentiation has no such surprises. Robert P.S. Analytically, we might say that integrals are easier to reason about. They're much more well-behaved than derivatives. On Fri, Dec 16, 2016 at 1:54 PM, Dan Asimov <asimov@msri.org> wrote:
Why is it usually so easy to differentiate a function defined by an exact formula, but so much more difficult to integrate?
If this question can be made rigorous, how might that be done?
(And if so, what is the rigorous answer, or at least a method of approaching it?)
—Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun