From: Dido Sevilla <dido.sevilla@...>
Date: 2009-02-08T01:04:45+09:00
Subject: Re: Integration/Antidifferentiation Program

On Fri, Apr 28, 2006 at 1:31 AM, SleepJunkie <SleepJunk13@gmail.com> wrote:
> I'd like to write a program to integrate derivatives to save time on
> calc homework and get extra credit in the class. I'm not completely sure
> where to start.
>

http://en.wikipedia.org/wiki/Risch_algorithm

There's apparently an algorithm out there that can do symbolic
integration of any elementary function, and returns either its
antiderivative, or that its antiderivative is not expressible in terms
of elementary functions. The only problem that it requires at several
points to determine whether two expressions are equal, and that,
unfortunately, is difficult, and in some cases actually impossible
(undecidable) depending on what functions you allow. See

http://en.wikipedia.org/wiki/Richardson's_theorem

There's also a whole theory built up from this called differential
Galois theory, which extends ideas originally developed by Évariste
Galois for the solvability of polynomial equations into solving
differential equations, which is really what antidifferentiation is
all about:

http://en.wikipedia.org/wiki/Differential_Galois_theory

You're treading in deep water here, brother. Beware.

-- 
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