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Differentiation under the integral sign needs a theorem with a proof. There are several versions of the theorem. The best proofs I know of are from the domina

Differentiation under the integral sign needs a theorem with a proof. There are several versions of the theorem. The best proofs I know of are from the dominated convergence theorem of measure theory. For the Putnam exam, yes, it's challenging. It's so challenging that doing the work to do well on the exam is comparable with creating and publishing peer reviewed original research in math, and such a publication will likely do much more for the academic progress for a student than the Putnam. Indeed, soon enough in math graduate school at a good research university, a student learns that everyone admits that no one can carry all the library around between their ears and doesn't try to. Instead, the three most important measures of progress are, in no particular order, research, research, and research. Finally, IIRC, how to do integration of algebraic expressions in closed form has long been a solved problem and long ago was coded up in some software -- IIRC, the software will report the result or state that no closed form solution exists.

imurray 8y ago
Finally, IIRC, how to do integration of algebraic expressions in closed form has long been a solved problem and long ago was coded up in some software -- IIRC, the software will report the result or state that no closed form solution exists. Are you thinking of indefinite integrals? https://en.wikipedia.org/wiki/Risch_algorithm https://en.wikipedia.org/wiki/Risch_algorithm This article is about a trick for definite integrals. If you can do the indefinite integral you can do a definite one, but not necessarily vice versa.
- graycat 8y ago
  You are no doubt ahead of me on this. With a little review (the Internet, Google, and Wikipedia are a new world), I recall that I was thinking of Macsyma with apparently a nice description at https://en.wikipedia.org/wiki/Macsyma https://en.wikipedia.org/wiki/Macsyma Apparently the software is still available, likely better than the original which apparently goes back to the MIT project MAC. The Wikipedia description early on mentions indefinite integration. IIRC the software implements algorithms that "solve" the problem of indefinite integration, calculus course "anti-differentiation". I'm plainly passing out what I remember, some of it confirmed by the Wikipedia page, and memory of hearsay. But likely now could download a copy of the software with whatever documentation they have and see how good it is, how much it claims, and that's a lot more than I could do the last time I heard anything about Macsyma. I'm not going to set my startup aside and rush to have fun with Macsyma, as much fun as I have had with calculus, but I will make a note in my little system for such chunks of information!

imurray 8y ago

Finally, IIRC, how to do integration of algebraic expressions in closed form has long been a solved problem and long ago was coded up in some software -- IIRC, the software will report the result or state that no closed form solution exists. Are you thinking of indefinite integrals? https://en.wikipedia.org/wiki/Risch_algorithm https://en.wikipedia.org/wiki/Risch_algorithm This article is about a trick for definite integrals. If you can do the indefinite integral you can do a definite one, but not necessarily vice versa.

graycat 8y ago
You are no doubt ahead of me on this. With a little review (the Internet, Google, and Wikipedia are a new world), I recall that I was thinking of Macsyma with apparently a nice description at https://en.wikipedia.org/wiki/Macsyma https://en.wikipedia.org/wiki/Macsyma Apparently the software is still available, likely better than the original which apparently goes back to the MIT project MAC. The Wikipedia description early on mentions indefinite integration. IIRC the software implements algorithms that "solve" the problem of indefinite integration, calculus course "anti-differentiation". I'm plainly passing out what I remember, some of it confirmed by the Wikipedia page, and memory of hearsay. But likely now could download a copy of the software with whatever documentation they have and see how good it is, how much it claims, and that's a lot more than I could do the last time I heard anything about Macsyma. I'm not going to set my startup aside and rush to have fun with Macsyma, as much fun as I have had with calculus, but I will make a note in my little system for such chunks of information!