Abstract: The numerical approximation of integrals of the form $$\int\limits_0^1 {{t^a}{{\left( {1 - t} \right)}^b}F\left( t \right)dt}$$ (1) have been considered by many authors under the assumption that the integrals converge. In this paper however, we shall assume that both a and b are real and non-integral and that either a <-1 or b <-1 or both (that is, the integrals in (1) are divergent. See also, for example, KUTT [5] or NINHAM [7]).