Adaptive integration of functions maintaining a constant sign of the sixth derivative

This paper introduces an adaptive numerical integration method specifically designed for functions of class C⁶[a,b] with a sixth derivative of constant sign. The approach is based on a refined inequality involving the three-point Gauss quadrature ( G ) and the four-point Lobatto quadrature ( L ). We utilize a specific linear combination, Q = 3/4G + 1/4L , and demonstrate that for functions with a sixth derivative of constant sign, the approximation error is effectively controlled by the difference between these two quadratures. We provide a rigorous justification for the stopping criterion of the adaptive algorithm. Numerical experiments, including the approximation of the integrals of a reciprocal function as well as the exponential function, show that the proposed method significantly outperforms existing adaptive techniques designed for lower-order convexity, requiring substantially fewer subintervals to achieve high precision (up to 10⁻¹⁶ ).