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.