Through some questionable methods, there lies an answer to the following integral.
$$\int_{-a}^a\frac{dx}x=0$$
You may question its soundness at first glance since:
$$\int_{-a}^a\frac{dx}x=\int_{0}^a\frac{dx}x+\int_{-a}^0\frac{dx}x=\infty-\infty?$$
But graphically, it works out well.
You may find similar examples of Cauchy principal values. They assign values to divergent definite integrals.
Similarly, you can assign values to divergent summations:
$$1+2+3+4+\dots=-\frac1{12}$$
You may find methods to evaluate such a summation here.
And while such things shouldn't be able to be defined, it turns out that we can evaluate these weird things using special methods or, sometimes, using the regular methods, methods we usually place restriction on.
But then, what's the point? What use does a Cauchy principal value and divergent summation have? I can't find too much (actually, I can't find any) information concerning the usage of such values/answers and would like to know any, no matter how small, use of such principal values and divergent summations, even if the application is another math related idea.
A main application is given precisely in link that you included to wikipedia. Although this reference (to wikipedia) is irrelevant for any serious discussion (either by me or you), principal values do play a role in distribution theory (others may call it "theory of generalized functions", which is not quite appropriate).
True that distribution theory failed to solve the problems that it initially proposed to solve, in connection to certain classes of partial differential equations, or in computing rigorously some series with physical applications. But it did give a sound ground for a rigorous playground on important generalizations of what you call "regular methods".
It is in particular quite welcome that principal values allow to define a very explicit class of distributions.