ca.analysis and odes - An integral arising in statistics

The integral I need:
$$t(x)=int_{-K}^{K}frac{exp(ixy)}{1+y^{2q}}dy$$

$K<infty$, q natural number

For q=1 this integral is
$$pi/2-int_{Arc}frac{exp(ixy)}{1+y^{2}}dy$$
Where Arc has radius $K$

Upper bound is $$Kpi/(K^2-1)^2$$

Can I obtain a better expression for the integral?

One more question about this integral. For K<1 this integral is just
$$-int_{Arc}frac{exp(ixy)}{1+y^{2}}dy?$$

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