An analysis of a method for the numerical evaluation of the integral \(\int\limits_a^b { f\left( x \right)} dx\) is presented. The method introduces a change of variable, x = x(q), with the property that dnx/dqnis zero at x = a, x = b for n = 0, 1, 2,... N, where N is an integer to be chosen. The Euler-Maclaurin formula shows that the resulting integral in the variable q is ideally suited for numerical integration, using equally spaced points and equal weights in q-space. Examples are given for various integrals which occur in quantum chemistry and applications to more than one dimension are discussed.