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From the Fun Fact files, here is a Random 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 using integration by parts in calculus. 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." Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

The Link for this Fun Fact: is directly accessible here.

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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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