Given the integral:
$I = \int_0^a{e^{-\lambda g(x)}f(x)dx}$ Where $g(x)$ and $f(x$) are both low order positive polynomials, and $\lambda \gg 1$, Laplace's method is commonly used to approximate the integral by using the first or second derivatives of $g(x)$.
Now assuming we know everything about $f(x)$, but do not know $\lambda$. Assume also for simplicity that $g(x)=x$. Is there a way of expressing the integral $I$ only in terms of: $G = \int_0^a{e^{-\lambda g(x)}dx}$ and derivatives of the function $f(x)$? I've tried expanding $f(x)$ as a Taylor series, but after the first term, I end up with a denominator containing $\lambda$ which is not known. Ideally I'd like to have something like: $I \approx f(0)G + f'(0)\times(...)$ With no $\lambda$ dependence outside of $G$.
Is this possible?