$$\frac{1}{(2\pi)^n} \int e^{i\xi\cdot x}e^{-|\xi|}d\xi= \frac{\Gamma((n+1)/2)}{\pi^{(n+1)/2}}\frac{1}{(|x|^2+1)^{(n+1)/2}}.$$

$$\frac{1}{(2\pi)^n} \int e^{i\xi\cdot x}e^{-|\xi|}d\xi= \frac{\Gamma((n+1)/2)}{\pi^{(n+1)/2}}\frac{1}{(|x|^2+1)^{(n+1)/2}}.$$ Hint: up to a constant factor, the RHS is the value of the Poisson kernel for the upper half-space at the point $(0,0,...0,1)$. The LHS is what you obtain when you solve the Laplace equation for the upper half-space by Fourier method.