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We present a novel numerical integration technique, Neural Network Integration, or NNI, where shallow neural network design is used to approximate an integrand function within a bounded set. This function approximation is such that a closed-form solution exists to its definite integral across any generalized polyhedron within the network's domain. This closed-form solution allows for fast integral evaluation of the function across different bounds, following the initial training of the network. In other words, it becomes possible to “pre-compute” the numerical integration problem, allowing for rapid evaluation later. Experimental tests are performed using the Genz integration test functions. These experiments show NNI to be a viable integration method, working best on predictable integrand functions, but worse results on singular and non-smooth functions. NNI is proposed as a solution to problems where numerical integrations of higher dimension must be performed over different domains frequently or rapidly and with low memory requirements, such as in real-time or embedded engineering applications. The application of this method to the optimization of integral functions is also discussed.