Talk:2117: Differentiation and Integration
- Calculus. 188.8.131.52 18:16, 27 February 2019 (UTC)
Basically, integration is easy to do by hand, but integration, even of things that look simple on paper, can be very difficult, as well as easy to mess up or get lost in.
And Calc 2 is why I stopped being a Computer Science major and moved (eventually) to majoring in English. Consistent 4.0s in math through Trig and Calc I ... 1.6 in Calc II, retook and got a 1.8. Without the Calc, couldn't do the physics; without the physics, couldn't get my 2-yr degree and move on from community college to a full university. I don't know what all the integration stuff in the flowchart is (since I didn't do well in Calc and it was a long time ago), but there's so very many things that become nonelementary integrals that all sorts of special tricks have to be employed for things that look like they should be easy. It's like having a problem that's very easy to do division on, but requires special advanced mathematical tricks to use multiplication upon.184.108.40.206 19:07, 27 February 2019 (UTC)
Basic ideas: Integration by parts is the reverse of the Product Rule. Substitution is the reverse of the Chain Rule. Cauchy's Formula gives the result of a contour integration in the complex plane, using "singularities" of the integrand. Partial fractions is just splitting up one complex fraction into a sum of simple fractions, which is relevant because they are easier to integrate. Stokes theorem is the relationship between an integral over an area, and an integral over the boundary of said area. Riemann integration was the first rigorous definition of integration. This has been superseded by Lesbesgue integration. Bessel functions are like 2d versions of sin and cos, and turn up sometimes when doing integration.220.127.116.11 20:14, 27 February 2019 (UTC)
Shouldn't Wolfram Alpha be somewhere in that flowchart? 18.104.22.168 20:54, 27 February 2019 (UTC)
Glad to see I'm not the only one who is too dumb to integrate 22.214.171.124 21:02, 27 February 2019 (UTC)
- Symbolic differentiation is just going through algorithm ; there are few functions which don't have it but they tend to be constructed in complicated way, and if function have differentiation it's usually easy to find it. Symbolic integration requires lot of thinking and trial and error ; even very easy function may lack primitive function and even if they don't, you may be unable to find it except randomly. If it's exercise in book, the ones for differentiation are done by thinking about some interesting function and putting it there. The ones for integration are done by thinking about some interesting function and putting it's differentiation there. -- Hkmaly (talk) 23:38, 27 February 2019 (UTC)
Oddly enough it mentions Riemann integration, but that is the integral most people know how to use. Turns out there are a lot more (e.g. lebesgue and generalized riemann integrals). I'm halfway through a second semester of real analysis and was floored by how involved integration can be. 126.96.36.199 21:36, 27 February 2019 (UTC)
One of my professors once said: "Never try to integrate a function. Almost all (in a strict mathematical sense) functions are impossible to integrate, so there is no reason why you should even try." --188.8.131.52 07:52, 28 February 2019 (UTC)
I thought I could contribute to the article with a better explanation of the Risch algorithm, since I have a bit of expertise here -- I've read all the original papers, plus the Cherry papers that add the extra features like Li and erf. I pulled out some of the old papers to review my knowledge of symbolic differential algebra (it's been a while!) then typed up a careful explanation which corrected some errors in the original description and fleshed out many more details... possibly excessively, but hey, that's kind of our calling here.
Then I saw that Glassvein completely removed my version for what appears to be the original without so much as a mention in the edit description. What gives? I