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Mathematica and Maple disagree on this integral

• To: mathgroup at smc.vnet.net
• Subject: [mg33091] Mathematica and Maple disagree on this integral
• From: Ben Crain <bcrain at bellatlantic.net>
• Date: Fri, 1 Mar 2002 06:52:16 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

What is the definite integral of Sec(x), from 0 to pi?  A textbook
answer (Stewart, Calculus) is that it diverges.
And that is the answer Maple gives (calling it "undefined").  But
Mathematica returns 0.

The integral is split into two improper integrals, from 0 to pi/2 and
from pi/2 to pi.  Each, by itself, diverges.  The textbook definition
requires both improper integrals to separately converge for the total
integral to converge.  By that definition, Maple is right.  But does
that make sense?  The second improper integral is just the negative of
the first, and they exactly cancel out for the antiderivative
ln(abs(sec(t) + tan(t)) at any t close to pi/2.  Why don't they exactly
offset each other in the limit, as t goes to pi/2, and yield 0?  Why
shouldn't the integral be so defined, instead of the textbook
requirement that the improper integrals must separately converge.

• Follow-Ups:
  • Re: Mathematica and Maple disagree on this integral
    • From: "Philippe Dumas" <dumasphi@noos.fr>