Analogue of Newton–Cotes formulas for numerical integration of functions with a boundary-layer component

The numerical integration of functions with a boundary-layer component whose derivatives are not uniformly bounded is investigated. The Newton–Cotes formulas as applied to such functions can lead to significant errors. An analogue of Newton–Cotes formulas that is exact for the boundary-layer component is constructed. For the resulting formula, an error estimate that is uniform with respect to the boundary-layer component and its derivatives is obtained. Numerical results that agree with the error estimates are presented.