ProofEasily, note that \begin{equation*} \int_{-\infty}^{\infty} \frac{1}{1+x^2} dx = tan^{-1}(\infty) - tan{-1}(-\infty) = \pi/2 - (-\pi/2) = \pi. \end{equation*} Dividing by \(\pi\) gives the Cauchy probability function integrates to 1.