----- Original Message ----- From: "R. William Gosper" <rwg@osots.com> To: <math-fun@mailman.xmission.com> Sent: Saturday, November 25, 2006 18:14 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.
True. But the Wolfram Functions site gives, in essence, the result below as the second formula at <http://functions.wolfram.com/IntegerFunctions/Floor/21/02/01/>.
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
The CAS Derive is particularly adept at finding such antiderivatives. For example, it gives the result above very quickly. David