----- Original Message ----- From: "Daniel Asimov" <dasimov@earthlink.net> To: "math-fun" <math-fun@mailman.xmission.com> Sent: Tuesday, November 28, 2006 17:01 Subject: Re: [math-fun] integrate(x^p floor(x)^q,x)
Apologies to whoever posted this, whose identity I've forgotten:
<< When I participated in Maple beta program, I submitted this bug. It was a heated discussion there about it, and I was told by Maple developers that in Maple indefinite integrals are defined up to "a piecewise constant", i.e. integrals of continuous functions can be discontinuous. The bug was classified as "works as designed".
I find this philosophy rather ridiculous for antiderivatives of continuous functions.
OTOH there are continuous functions for which useful antiderivatives can be given "up to piecewise constants" but for which continuous antiderivatives cannot be given in closed form in terms of familiar functions. Here's an example due to Robert Israel: Consider the continuous function |exp(x) + cos(x)|. An antiderivative as given by Derive is sgn(exp(x) + cos(x))*(exp(x) + sin(x)) which has discontinuities at the zeros of exp(x) + cos(x). Now, if only we could express those zeros in closed form... but we can't. I think that it is better for a CAS to give an answer such as Derive's than to just return the indefinite integral unevaluated.
But for antiderivatives of functions continuous but on natural domains that are not connected, like the ubiquitous
f(x) = 1/x,
calculus books should give its antiderivative as
ln|x| + C_1, x > 0 F(x) = { ln|x| + C_2, x < 0
for arbitrary choice of constants C_1, C_2,
and likewise for 1/x^p, p > 0, p <> 1 (and mutatis mutandum for sec(x), csc(x), etc.).
I have a different viewpoint. If interested, see "Improper integrals made proper" (sci.math, 2006 May 30) <http://groups.google.com/group/sci.math/msg/a03ffa4b2ff03f5a>. David W. Cantrell