I think I'll adopt a modified version of Veit's suggestion and say "Remember why it's called an INDEFINITE integral." Jim On Thursday, March 13, 2014, Veit Elser <ve10@cornell.edu> wrote:
I would simply remind them that it is the derivatives of the indefinite integrals that have to be equal, not the indefinite integrals themselves. After all, they are "indefinite"?
Veit
On Mar 13, 2014, at 4:27 PM, "James Propp" <jamespropp@gmail.com<javascript:;>> wrote:
I just invented this paradox, though I doubt I'm the first:
"On the one hand, the indefinite integral of 2x+2 w.r.t. x is x^2 + 2x + C; on the other hand, putting u = x+1, we can write the integral as the integral of 2u w.r.t. u, which is u^2 + C = x^2 + 2x + 1 + C. Equating x^2 + 2x + C with x^2 + 2x + 1 + C we get 0 = 1."
Here's my question for you guys: For those students who are stymied by this paradox, what would be a good hint? I've tried "Think about what indefinite integration IS", but that just seems to confuse them more. Is there a good hint that doesn't give away the game?
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