Let [math]X[/math] be a continuous random variable whose characteristic function [math]k_X(\tau)[/math] is

[[math]] k_X(\tau) = e^{-|\tau|}, \qquad -\infty \lt \tau \lt +\infty\ . [[/math]]

Show directly that the density [math]f_X[/math] of [math]X[/math] is

[[math]] f_X(x) = \frac1{\pi(1 + x^2)}\ . [[/math]]