The author of my complex analysis textbook asks the reader to find the Cauchy principal value of absolutely convergent real-valued integrals such as $\displaystyle\int_{{\color{red}{-\infty}}}^\infty \frac{\cos x}{1+x^{2}} \ dx $.
For a long time I thought that meant that $\text{PV}\int_{-\infty}^\infty \frac{\cos x}{1+x^{2}} \ dx = \lim_{R \to \infty} \int_{-R}^{R} \frac{\cos x}{1+x^{2}} \ dx \ne \int_{-\infty}^\infty \frac{\cos x}{1+x^{2}} \ dx. $
But doesn't an absolutely convergent real-valued integral equal its Cauchy principal value?