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The corrected two-point weighted quadrature formulae

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 $[-1, 1]$. The 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.

Weight function; corrected quadrature formula; two-point quadrature formula; $w-$harmonic sequences of functions; $L_p$ spaces; corrected Gauss formula; weighted corrected Maclaurin formula; weighted corrected trapezoid formula; weighted corrected midpoint formula; weighted corrected Newton-Cotes two-point formula

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