I couldn't calculate $\int \limits_{-\infty}^{+\infty} \frac{e^{-itx}}{2\pi} \frac{1}{a^2+x^2} dx. $
I can either turn this into something along the lines of $\large \int \limits_0^{\pi/2} \cos( t \cdot \tan x ) dx$ or $ \large \int \limits_0^{+\infty} \frac{\cos tx}{a + ix} dx$ neither of which I can solve.
I've been told, that some tools of complex calculus could simplify this, but my book hasn't covered any before giving the exercise, so I wonder if there is a way without it.
Thanks.