We present a neural network approach based on the cosine basis functions to solve numerical integration. The convergence of neural networks algorithm and the theorem to solve numerical integration are also proven. To prompt neural network learning, we propose the momentum method, the conjugate gradient (CG) method, and the trancated Newton (TN) method for training neural networks. This article also gives examples of numerical integration based on neural networks and compares the advantages and disadvantages of each method. The results show that the numerical integration method presented here has a high accuracy and a fast convergence rate, hence, its application values is significant in the engineering practice.