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You need to include absolute value signs in the argument of the logarithm:
ln|x-1| from 0-1 and
ln|x-1| from 1-2
What you find is that both the limits diverge logarithmically, so the integral doesn't exist. However, if you add both the terms together and take the limit in one go, then the divergent terms cancel. This is called the Cauchy principal value of the integral.