On representation of certain integrals using the values of a function and its derivatives

The problem of integrating a function on the basis of its approximation by two-point Hermite interpolation polynomials is considered. Quadrature formulas are obtained for the general case, when the orders of the derivatives given at the endpoints of the segment can be not equal to each other. The formula for the remainder term is presented and the error of numerical integration is estimated. Examples of integrating functions with data on error and its estimation are given. A two-point approximation of the integrals is compared with a method based on the Euler-Maclaurin formula. Comparison of the two-point integration method with the approach based on the use of the Euler-Maclaurin formula showed that for sufficiently smooth functions the accuracy of two-point integration is significantly higher than by the Euler-Maclaurin formula. An example of an integral is given for which its approximations obtained using the Euler-Maclaurin formula diverge, and those obtained by the formula two-point integration converge quickly enough. We also note that, in contrast to the Euler-Maclaurin formula, the two-point integration formula is also applicable in the case when the maximum orders of the derivatives at the ends of the integration interval may not be equal to each other, which is important in practical applications.