The corrected two-point weighted quadrature formulae

Abstract

We derive a corrected version of the weighted two-point quadrature formula, which provides a better approximation accuracy than the ordinary two-point quadrature formulae. In the corrected two-point formula the integral is approximated both with the values of the integrand in nodes −x and x, and the values of its first derivative at the endpoints of the interval (−1,1) .T he error estimates under various regularity conditions for such formulae are established. As special cases, the corrected two-point formulae of Gauss type are obtained. Also, corrected version of weighted trapezoid, midpoint, two-point Maclaurin and two-point Newton-Cotes formulae are considered.