Optimal quadrature formulas with derivative exact for trigonometric functions

In this article, we study how to find the coefficients of a quadrature formula with derivative using the ’ - function method. The ’ - function method allows for construction of optimal quadrature formulas in approximate. Also, the functional error was analyzed using the quadrature formula constructed using the ’-function, and the results were supported by precise mathematical expressions. With arbitrary fixed nodes the optimality conditions for the quadrature formula are analyzed and the method for determining the components of that formula are considered. The explicite forms for the coefficients of the optimal quadrature formula are obtained. In particular, for the equally spaced nodes the Euler-Maclaurin type formula is obtained.