Laplace's Method for Integral asymptotes when g(c) = 0 I am using these notes as my reference, but I am running into some questions.  
Say I am trying to find, for large $\lambda$
$$I(\lambda)=\int_0^{\pi/2}dxe^{-\lambda\sin^2(x)}$$
This has our maximum at $c=0$, where g(c)=0 and g'(c)$\neq$0.  So when I pull out the $e^{\lambda g(c)}$, that is just 1, so I continue to expand g(x) about x = 0, and then change the bounds from -Infinity to Infinity.
$$I(\lambda)\approx\int_{-\infty}^\infty dx\exp[-\lambda(x^2-x^4/3+...)]$$
To first order, this is just $\sqrt{(\pi/x)}$, but the second order diverges.
The reason this stumps me is because I can find an exact solution that depends on $e^x$ so I am not sure where i am going wrong in computing the behavior.

Note that you have an endpoint maximum, and hence the integral should only be one-sided; this is the sort of case where Watson's lemma applies.
