PlusDistribution[1/(1 - x)] denotes a distribution (in
the sense of the “+” prescription).
PlusDistribution[1/(1 - x)]\left(\frac{1}{1-x}\right)_+
PlusDistribution[Log[1 - x]/(1 - x)]\left(\frac{\log (1-x)}{1-x}\right)_+
Integrate2[PlusDistribution[1/(1 - x)], {x, 0, 1}]0
Integrate2[PlusDistribution[Log[1 - x]/(1 - x)], {x, 0, 1}]0
Integrate2[PlusDistribution[Log[1 - x]^2/(1 - x)], {x, 0, 1}]0