Approximating rapidly divergent integrals

A while back I ran across a paper [1] giving a trick for evaluating integrals of the form

where M is large and f is an increasing function. For large M, the integral is asymptotically

That is, the ratio of A(M) to I(M) goes to 1 as M goes to infinity.

This looks like a strange variation on Laplace's approximation. And although Laplace's method is often useful in practice, no applications of the approximation above come to mind. Any ideas? I have a vague feeling I could have used something like this before.

There is one more requirement on the function f. In addition to being an increasing function, it must also satisfy

In [1] the author gives several examples, including using f(x) = x^2. If we wanted to approximate

the method above gives

exp(10000)/200 = 4.4034 * 10⁴³⁴⁰

whereas the correct value to five significant figures is 4.4036 * 10⁴³⁴⁰.

Even getting an estimate of the order of magnitude for such a large integral could be useful, and the approximation does better than that.

[1] Ira Rosenholtz. Estimating Large Integrals: The Bigger They Are, The Harder They Fall. The College Mathematics Journal, Vol. 32, No. 5 (Nov., 2001), pp. 322-329