How do I put $g(x,y)$ into the second integration in $\int_{-\infty}^\infty g(x,y)(\int_{-\infty}^\infty f(x,y)\mathop{\mathrm dy})\mathrm dx $

Question

We have $$\int_{-\infty}^\infty g(x,y)\left(\int_{-\infty}^\infty f(x,y)\mathop{\mathrm dy}\right)\mathrm dx $$

What I would like to have is to put $g(x,y)$ into the second integral, so that it is easier to compute, when I am not able to calculate the integral from f itself.

If it is not possible, then maybe some of you know what I might do in this particular case, when $g(x,t)=e^{tx}$ and $f(x,t)=e^{-|t|^{\alpha}}e^{-itx}$, if one is interested this is a function generating moments for Alpha stable distributions.

Any help is appreciated, thanks!