  In this paper we present a new family of rules for numerical integration.
This family has up to half the error of the widely used Newton-Cotes rules when
a sufficient number of points is evaluated and also much better numerical
stability for high orders. These rules can be written as the midpoint rule with
a correction term, providing a straightforward and computationally cheap way to
obtain error estimations. The rules are interpolatory and use evenly spaced
points, which makes them well suited for many practical applications. Their
major potential disadvantage is the use of points outside the integration
interval.
