Recently there was an excellent xkcd about differentiation and integration, see here.
This brings up thoughts on diff and int:
1) For some students Integration is when math gets hard.
Diff (at least on the level of Calc I) is rote memorization. A computer program can EASILY do diff
Integration by humans requires more guesswork and thought, Computers can now do it very well but I think that it was harder to get to work.
Someone who has worked on programs for both, please comment.
2) When I took Honors Calculus back in 1976 (from Jeff Cheeger at SUNY Stonybrook) he made a comment which really puzzled the class, and myself, but later I understood it:
Integration is easier than Differentiation
The class thought this was very odd since the problem of, GIVEN a function, find its diff was easier than GIVEN a function, find its int. And of course I am talking about the kinds of functions one is
given in Calc I and Calc II, so this is not meant to be a formal statement.
What he meant was that integration has better mathematical properties than differentiation. For example, differentiating the function f(x)=abs(x) (absolute value of x) is problematic at 0, where it has no problem with integration anywhere (alas, if only our society was as relaxed about integration as f(x)=abs(x) is).
So I would say that the class and Dr. Cheeger were both right (someone else might say they were both wrong) we were just looking at different notions of easy and hard.
Are there other cases in math where `easy' and `hard' can mean very different things?