I wonder if there is any sensible generalization of regularization which would be able to ascribe finite values to $\int_0^\infty \tan x \,dx$ and $\int_{-\infty}^0 \psi(x)dx$?

Perticularly, since $\tan x$ starts with a positive segment, $\int_0^\infty \tan x\, dx$ logically should be either positive or at least non-negative (compare $\int_0^\infty \sin x\, dx$ which sums up to $1$ using Cesaro integration).

On the other hand the fact that the centers of mass of positive and negative segments coincide in each period, speaks in favor of the natural regularization being zero.

Using enough abstract nonsense, one can integrate arbitrary functions, but in a rather useless way.$\endgroup$