I have encountered recently several integrals of the form
$$ I(k)=\int\limits_{-\infty}^\infty dx \ \sin(kx)f(x)g\left(\sqrt{x^2+m^2}\right) $$
Where $m$ is a real constant, and $f(x)\sim 1/x$ as $x\to 0$, but elsewhere is infinitely differentiable. $g(x) \sim x^{-n}e^{-x}$ as $x \to \infty$ for some $n>0$ and is infinitely differentiable. Is it possible to say anything about the large $k$ behavior of $I(k)$? Perhaps in terms of the derivatives of $f$ or $g$?
So far what I have done is make the replacement $\sqrt{x^2+m^2}\sim m+x^2/2m$ as $x\to 0$ in the argument of $g$, and evaluate the resulting integrals of the form
$$ \int\limits_{-\infty}^\infty dx \ \sin(kx)f(x)g\left(m+x^2/2m \right) $$
on a case-by-case basis. This feels a little unnecessary, as I am only interested in the leading order behavior, but I am unsure of a better way to proceed.
Happy to provide more information if necessary.