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". A workaround for that (in Maple) is to use definite integral from 0 to x instead of indefinite integral. That also doesn't always work (sometimes it returns unevaluated), but as far as I recall, that gives the correct answer in this example. Alec Mihailovs http://mihailovs.com/Alec/ ----- Original Message ----- From: "R. William Gosper" <rwg@osots.com> To: <math-fun@mailman.xmission.com> Sent: Saturday, November 25, 2006 12:14 PM Subject: [math-fun] integrate(x^p floor(x)^q,x)
Maple and Macsyma incorrectly give x floor x for integrate(floor(x),x). (Integrals are continuous.) Mma abstains. The answer is
ceiling(x) floor(x) + 1 (x - ----------) (ceiling(x) - 1) = floor(x) (x - ------------), 2 2
(products of discontinuous functions). More generally, one can find (e.g., by undetermined coefficients) a polynomial(floor x) of degree p+q which, when added to x^p floor(x)^q, renders the sum continuous. E.g.,
/ 4 4 8 7 [ 3 4 x floor (x) floor (x) 2 floor (x) I x floor (x) dx = ------------ - --------- - ----------- ] 4 8 7 /
5 3 3 floor (x) floor (x) 23 floor(x) + ----------- - --------- + -----------. 10 6 840
How does one seek in EIS a two-parameter family of polynomials with rational coefficents (presumably named Smarandache)? --rwg UNDERAGES UNGREASED DUNGAREES
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