I wonder what would be the asymptotic behaviour of $Re \int_0^{\infty}\,J_0(ks)e^{-i\sqrt{\alpha k}t}kdk$ for $s,\alpha$ large enough. $J_0$ is the Bessel function of zeroth order.
I feel it is strange problem. $f(x)$ grows as $x \rightarrow \infty$.
$f(k) = J_0(kr)k$ and $J_0 \sim k^{-1/2}$, so $f(k) \sim k^{1/2}$.
From other hand, $J_0 \sim \cos(ks) e^{-i \sqrt{\alpha k} t}$ product of two functions oscillating with different frequencies could give zero result, but $\int |f(k)| dk = \infty$.