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© 1999-2007 by Francis Edward Su
All rights reserved.

From the Fun Fact files, here is a Fun Fact at the Easy level:

One Equals Zero: Integral Form

Consider the following integral:


INTEGRAL (1/x) dx

Perform integration by parts: let

u = 1/x , dv = dx
du = -1/x2 dx , v = x

Then obtain:

INTEGRAL (1/x) dx = (1/x)*x - INTEGRAL x (-1/x2) dx
= 1 + INTEGRAL (1/x) dx

which implies that 0 = 1.

What's wrong with this calculation?

The Math Behind the Fact:
This is common mistake in integration by parts. Students often forget about the constant of integration for indefinite integrals. In this case, the constants on both sides will differ by 1.

How to Cite this Page:
Su, Francis E., et al. "One Equals Zero: Integral Form." Mudd Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

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References:
Keywords:    calculus, false proof, paradox
Subjects:    calculus, analysis, other
Level:    Easy
Fun Fact suggested by:   James Baglama
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