Two Formulas for Numerical Indefinite Integration

We derive two formulas for approximating the indefinite integral over a finite interval. The approximation error is $O({c^{ - c\sqrt m }})$ uniformly, where m is the number of integrand evaluations. The integrand is required to be analytic in the interior of the integration interval, but may be singular at the endpoints. Some sample calculations indicate that the actual convergence rate accords with the error bound derived.