Given an o-minimal structure R that expands the real ordered field, we want to investigate whether the expansion of R with a parametrized integral, whose integrand is definable in R, remains o-minimal. This has already been verified for the special instance of the real ordered field augmented by restricted analytic functions and power functions that have a real algebraic exponent. Additionally, it is also already known that the class of constructible functions is closed under parametrized integration. However, these results make use of preparation theorems and therefore do not give any insight into why parametrized integration in general is (most likely) a tame operation. We thus work with more general approaches that deal with parametrized integration directly. First, we want to lay out a way how to understand parametrized integration from a nonstandard perspective and then present some limit-existence results that are first indications that parametrized integration is indeed a tame operation.
