I've noticed that some complex analysis textbooks discuss evaluating real-valued integrals like $\int_{0}^{\infty} \frac{\sqrt{x}}{1+x^{2}} \, dx $ using a keyhole contour before they have defined the Cauchy principal value of an integral (first definition).
But isn't using a keyhole contour a principal value approach in the sense that the contour approaches the singularity at the origin in a symmetrical way?