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General closed 4-point quadrature formulae of Euler type

General closed 4-point quadrature formulae where the integral over [0, 1] is approximated with the values of the function in points: 0, x, 1-x and 1 are derived. As special cases, Simpson's formula, Simpson's 3/8 formula and the Gauss 2-point formula are recaptured. Next, general corrected closed 4-point quadrature formulae are derived, and the Lobatto 4-point, corrected Simpson's, corrected Simpson's 3/8 formula and the corrected Gauss 2-point formula are obtained as special cases. We call "corrected" such quadrature formulae where the integral is approximated not only with the values of the integrand at certain points but with the values of its first derivative at the end points of the interval as well.

quadrature formulae; corrected quadrature formulae; Lobatto 4-point formula; Bernoulli polynomials; extended Euler formulae

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