This monograph is a testimony of the impact over Computational Analysis of some new trigonometric and hyperbolic types of Taylor's formulae with integral remainders producing a rich collection of approximations of a very wide spectrum.This volume covers perturbed neural network approximations by themselves and with their connections to Brownian motion and stochastic processes, univariate and multivariate analytical inequalities (both ordinary and fractional), Korovkin theory, and approximations by singular integrals (both univariate and multivariate cases). These results are expected to find applications in the many areas of Pure and Applied Mathematics, Computer Science, Engineering, Artificial Intelligence, Machine Learning, Deep Learning, Analytical Inequalities, Approximation Theory, Statistics, Economics, amongst others. Thus, this treatise is suitable for researchers, graduate students, practitioners and seminars of related disciplines, and serves well as an invaluable resource for all Science and Engineering libraries.