Integration Formulas of Tenth Order and Higher

Abstract—Nowadays, due to the considerable growth of computer capacities, the development of more efficient quadrature formulas may seem unnecessary. However, if the calculation of each integrand value requires much computational time or we have to study the dependence of the integral on a large number of parameters the integrand is determined through, then it is necessary to use more efficient formulas. DOI: 10.3103/S0027132210040017 Quadrature formulas of 8th order for calculation of integrals of the form a∫ b f(x) dx are presented (and used) in reference-books and program packages. In fact, it is easy to calculate all the coefficients of quadra-ture formulas of Gauss type for the order 10 and higher. In particular, for “symmetric weights ” we have a∫ −a ρ(x)f(x) dx, where ρ(x) is an even weight function; the degree of the polynomial for the calculation of weights is decreased by two times, more precisely, P (x) = xθf(x2), where P (x) is the polynomial of degre