How Does Asymptotic Expansion Apply to Integrals with Oscillatory Behavior?

[Clausius2]

Hey guys,
I need help with the expansion of this integral:

$$\int_0^\infty Z(x) J_o(\lambda x)dx$$ for $$\lambda \rightarrow \infty$$

where I know that $$Z(x)\sim x^\sqrt{2}$$ for small $$x$$ and
exponentially small for large $$x$$

It seems with other examples that I have done that the major contribution to the integral comes from the region $$x\sim 1/\lambda$$. For larger $$x$$ the integrand oscillates rapidly and the integration cancels. One change of variable (re-scaling) that you may try is $$t=\lambda x$$. But if you do it you end up with a divergent integral. And at first glance the original integral is convergent. Any hints?

Thanks.

 

[Clausius2]

with divergent integral I mean that I arrive to this integral after the rescalement:

$$\frac{1}{\lambda^{\sqrt{2}+1}}\int_0^\infty t^{\sqrt{2}} J_o(t)dt$$