Challenges in Integrating the Fourier Transform of a Complex Function

[hamsterman]

I'm trying to find \frac{1}{2\pi}\int \limits_{-\infty}^{\infty}e^{-itx}\frac{1}{a^2+x^2}\mathrm{d}x where 'a' is a constant.

First I noticed that there is \frac {\partial \arctan x}{\partial x} in this and using a substitute got \int \limits_0^{\pi / 2}\cos( t \tan x )\mathrm{d}x with some constants in the gaps.
I then remember that I'm working in complex numbers, factored a^2+x^2 and got something essentially along the lines of \int \frac{e^x}{x}\mathrm{d}x, or maybe rather \int \limits_0^{\infty} \frac {\cos tx} {a - ix}\mathrm{d}x.

I can't integrate either.

 

[HallsofIvy]

Your integral does not contain "dx" or "dt". Without that we cannot tell what integration you intend. Is the problem
\int \frac{e^{-itx}}{a^2+ x^2} dx
or is it
\int \frac{e^{-itx}}{a^2+ x^2}dt
?

 

[hamsterman]

Oh, sorry. It's dx. I'll fix it right away.