  Many applications require the calculation of integrals of multidimensional
functions. A general and popular procedure is to estimate integrals by
averaging multiple evaluations of the function. Often, each evaluation of the
function entails costly computations. The use of a \emph{proxy} or surrogate
for the true function is useful if repeated evaluations are necessary. The
proxy is even more useful if its integral is known analytically and can be
calculated practically. We propose the use of a versatile yet simple class of
artificial neural networks -- sigmoidal universal approximators -- as a proxy
for functions whose integrals need to be estimated. We design a family of fixed
networks, which we call Q-NETs, that operate on parameters of a trained proxy
to calculate exact integrals over \emph{any subset of dimensions} of the input
domain. We identify transformations to the input space for which integrals may
be recalculated without resampling the integrand or retraining the proxy. We
highlight the benefits of this scheme for a few applications such as inverse
rendering, generation of procedural noise, visualization and simulation. The
proposed proxy is appealing in the following contexts: the dimensionality is
low ($<10$D); the estimation of integrals needs to be decoupled from the
sampling strategy; sparse, adaptive sampling is used; marginal functions need
to be known in functional form; or when powerful Single Instruction Multiple
Data/Thread (SIMD/SIMT) pipelines are available for computation.
