I think you need an i in there huh? Like:
Differentiate under the integral sign and use Cauchy's Theorem to express the Airy integral in terms of a contour integral over a square region bordering the real axis of height say i. Then since the integrand is analytic, we can express the Airy integral in terms of the horizontal leg at since the integral over the two vertical legs tend to zero. Then:
but that integrand is the differential of and since the integrand is analytic, we can just evaluate it at the endpoints:
and note if you substitute into , it's order is so we have:
which is zero.