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where the ellipsis (· · ·) indicates further terms in the various binomial expansions; we
keep as many as required in order to demonstrate that vanishes identically (at this
order), to leave

Thus we have found that

as as far as terms at here we see that the integration over x =O(l)
provides the dominant contribution to this value.

This example has presented, via a fairly routine calculation, the essential idea that
underpins this method for evaluating integrals. Of course, there is no need to exploit
this technique if the integral can be evaluated directly (as was the case here); let us
therefore examine another problem which is less elementary.

E2.6 Another integral
We wish to evaluate the integral

as here, the expansion of the integrand requires three different asymptotic
expansions (valid for x = O(l), Thus we obtain