Integrate[1/(x - 1), {x, 0, 3}, PrincipalValue -> True]
We have singularity at point "1".Then from wikipedia:
int = Integrate[1/(x - 1), {x, 0, 1 - e}, Assumptions -> e >= 0] + Integrate[1/(x - 1), {x, 1 + e, 3}, Assumptions -> e >= 0] Limit[int[[1]], e -> 0, Direction -> "FromAbove"] (* Log[2] *)
