Optimal quadrature rule with derivative in Sobolev space

As a result of extensive scientific research conducted on a global scale, approximating exact integrals and integral equations in the problems of the mechanics of liquids and gases with high accuracy leads to the construction of optimal quadrature formulas. Use of simple interpolation quadrature formulas in solving such problems requires a large amount of computational work. The effectiveness of quadrature formulas is characterized by their degree of accuracy and order of accuracy. In this work, the optimal quadrature formula with derivative is constructed using the first and second order derivatives of the function at the nodes in the real-valued Sobolev space.