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 . 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. <https://www.math.hmc.edu/funfacts>.

References:
R. Vakil, A Mathematical Mosaic, 1996. p. 199.

Fun Fact suggested by:
Joshua Sabloff

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