cv.complex variables - An integral arising in statistics(2)

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

$K<infty$, q natural number

For q=1 one can use contour integration.
So for K>1 we have :

$$pi/2-int_{Arc}frac{exp(ixy)}{1+y^{2}}dy $$
Where Arc has radius $K$

Is it correct that for K<1 this integral is:
$$-int_{Arc}frac{exp(ixy)}{1+y^{2}}dy ?$$

What about K=1?