Method of Steepest descent integral I am looking to evaluate the following asymptotic integral: 
Find the leading term of asymptotics as $\lambda\to\infty$
$$I(\lambda)=\int_0^1\cos(\lambda x^3)dx$$
Using method of steepest descent along a certain contour. I am having trouble approaching this problem as I don't understand it well. Any help would be appreciated.

To start, recognize that
$$
I(\lambda) = \operatorname{Re} \int_0^1 e^{i\lambda x^3}\,dx.
$$
Now there are several questions that you can ask to get yourself going:


*

*Where is the saddle point?

*What are the paths of steepest descent away from the saddle point?

*How can I deform my contour so that it follows this path of steepest descent?
The last one is a bit tricky since the endpoints of the contour are finite.  You only need to follow a portion of the path of steepest descent though; you can have the contour return to its start/endpoint afterwards.
Let me know if you get stuck on any of these.
