Integral inequalities prove extremely effective in obtaining error bounds for numerical integration formulas, which are particularly useful in optimization problems. This study presents reduced results of perturbed trapezoidal inequalities derived using the triangle inequality, Hölder’s inequality, and power mean inequalities for previously constructed nth-order differentiable s-convex and tgs-convex functions. These results help determine error limits for the trapezoidal rule and the remaining term of the midpoint formula in numerical integration. The study shows that these reduced inequalities yield better error limits. Additionally, an example involving a tgs-convex function defined under certain conditions demonstrates the practical application of these theoretical results.