Originally Posted by

**Guy** At least for someone who hasn't taken any complex analysis and doesn't know any beyond the really basic stuff, is there an easy way to do the following integral?

$\displaystyle \displaystyle \int_{\mathbb R} \frac 1 \pi \frac{e^{itx}}{1 + x^2} \mbox{ } dx = e^{-|t|} $

I trudged up a proof, but it uses techniques that I don't know. Obviously it is equivalent to evaluating

$\displaystyle \int_{\mathbb R} \frac 1 \pi \frac{\cos(tx)}{1 + x^2} \mbox{ } dx$

but I get the feeling looking at the result that this is something you would want to evaluate using the first expression.