I am trying to solve the following definite integral:

$$ \int\_{-\infty}^{+\infty} \frac{ \sin^2(\sqrt{(x-a)^2 + b^2}\,\,t)}{(x-a)^2 + b^2} dx$$
with $a$ and $b$ being real constants. Notice that the integrand is non-negative for all x , with a peak around $x=a$ for small values of $b/a$. So the integration does give a finite value. 

To go about this, I tried substituting $(x-a)^2 + b^2 = y^2$

This gives me $dx \,(x - a) = y \, dy$ whence:

$$ \int\_{?}^{+\infty} \frac{ \sin^2(|y|\,\,t)}{y} \frac{1}{\sqrt{y^2 - b^2}} dy$$

While the upper limit of y transforms to $+\infty$, clearly the lower limit is not $-\infty$ (even if it is $-\infty$ the integrand is odd so it evaluates to zero which cannot be true).

This suggests its a case of bad substitution. The only better alternative I can think of is to substitute $x-a = y$, this gives me:

$$ \int\_{-\infty}^{+\infty} \frac{ \sin^2(\sqrt{y^2 + b^2}\,\,t)}{y^2 + b^2} dy $$

Any thoughts on how I can get an expression for this integral ?

Thanks for your time!