I would like to find an asymptotic expansion of :
$$\int_0^x e^{t^2} \mathrm{d}t \text{ as } x \to \infty$$
I am enable to even find the first term. I tried to find a function $g$ such that : $g \sim e^{t^2}$ and such that $g$ is an easy integrable function yet $2te^{t^2}$ doesn't work so I am kind of lost.