I've been stuck on this one homework problem for nearly a day now, so I'd be really thankful for any pointers. The problem is to find
$$\int\int\int\_{\mathbb{R}^3} \frac{1}{\mathbf{x}^2+1}e^{-2\pi i (x\_1\zeta\_1+x\_2\zeta\_2+x\_3\zeta\_3)}dx\_1dx\_2dx\_3$$
where $\mathbf{x}\in \mathbb{R}^3$. I'm not sure how to go about this; I know that the Fourier transform of $1/(1+x^2)$ in $\mathbb{R}$ is $\pi e^{-2\pi |\zeta|}$, but I don't think that helps at all. I've tried changing it to spherical coordinates as well.

Also, if anyone could briefly explain how this relates to solving the equation $u(x)-\Delta u(x)=f(x)$, $x\in \mathbb{R}^3$ and $\Delta$ being the Laplacian operator, I'd be eternally grateful.