inverse fourier transform of exponencial

Question

Show that $F^{-1}(e^{-|x|}) =(\sqrt{2}/\sqrt{\pi})*1/(1+x^2)$ on $\mathbb R$. $F^{-1}$ is the inverse Fourier transform. Any help? how do you solve the integrals?

Answer 1

Hint: You have to split the integral:

$$\mathcal{F}^{-1}\{e^{-|x|}\}=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-|x|}e^{ixt}dx= \frac{1}{\sqrt{2\pi}}\left[ \int_{0}^{\infty}e^{-x(1-it)}dx+\int_{-\infty}^{0}e^{x(1+it)}dx\right]$$

I'm sure you can take it from here.