Quadrature formulae of Gauss type based on Euler identities

The aim of this paper is to derive quadrature formulae of Gauss type based on Euler identities. First, we derive quadrature formulae where the integral over [0,1] is approximated by values of the function in three points: x,1/2x,1/2 and 1−x1−x. As special cases, the Gauss 2-point formula, Simpson’s formula, dual Simpson’s formula and Maclaurin’s formula are obtained. Next, corrected Gauss 2-point formulae are derived and finally, the Gauss 3-point formulae and the corrected Gauss 3-point formulae are considered. We call “corrected” such quadrature formulae where the integral is approximated both with the values of the integrand in certain points and the values of its first derivative in the end points of the interval. Corrected formulae have a degree of exactness higher than the adjoint original formulae.