Determining Cauchy principal value of divergent integrals

Homework Statement

So I've found a ton of examples that show you how to find cauchy principal values of convergent integrals because it is just equal to the value of that integral and you prove that the semi-circle contribution goes to zero. However, I need to find some Cauchy principal values of divergent integrals and I can't find any examples or even problems in any books that have these. Perhaps I'm just looking in the wrong place I'm not sure. If anyone could point me in the direction of any examples where principal values are found of divergent integrals that would be amazing, thank you.

Homework Equations

$$P.V. \int^{\infty}_{-\infty}\frac{sin2xdx}{x+4}$$
$$P.V.\int^{\infty}_{-\infty}\frac{cos2xdx}{x^{2}-16}$$

The Attempt at a Solution

I've gone through many examples but they just prove that the contribution of the arc on the contour goes to zero and so the principal value equals the value of the convergent integral. Any examples resembling my problems would be great.[/B]