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I got an exam in calculus in a few weeks, and lots of questions coming up. Here's one of them:

We learned that the Fourier Transform of

[tex] f(x) = e^{-|x|} [/tex]

is

[tex] \hat f(\omega) = \sqrt{\frac{2}{\pi}}\frac{1}{1+\omega^2}[/tex]

I've got no problem with this one. Now, since [tex] \hat f(\omega)[/tex] is Lebesgue - integrable, the inverse Fourier transform exists and should be

[tex] \hat \hat f(-x) = e^{-|x|} [/tex]

To show this the 'hard way', I want to calculate the integral

[tex] \sqrt{\frac{1}{2 \pi}} \int \limits_{-\infty}^\infty \frac{1}{1+t^2} e^{i \omega t} \mathrm{d} t [/tex]

Well, I just don't know how to do this one. Partial Integration doesn't seem to work, and I can't find neither a good substition nor a clever use of Fubini's Theorem.

Would be thankful for any help.