How to compute this integral in general case? $$t(x)=\int_{-\infty}^{\infty}\frac{\exp(ixy)}{1+y^{2q}}dy$$

Mathematica can compute it when q is known. For example,for q=1 this integral is $$\exp(-{\left|x\right|})\pi$$

But even in this case, I don't really know how to get this result.