What is wrong with this proof, by integration by parts, that 0 = 1?

Problem

What is wrong with this proof, by integration by parts, that 0 = 1?

If you do not know calculus - stop here.

I will use { to indicate an integral sign.

We all learned that {udv = uv - {vdu
To avoid generating an obvious solution of an integrating constant,
I will integrate (1/x)dx from a to b
let 1/x = u and dx = v then du = -1/x**2 dx and v = x

a{b (1/x)dx = (1/x)*x - a{b x*(-1/x**2) dx
or
a{b (1/x)dx = 1 + a{b (1/x) dx
hence 0 = 1

-- Larry Fishbach <maven@hercules.lerc.nasa.gov> (January 1995)

[ Editor: the posting's subject was "A harder to see fallacy of 0=1". ]