  Conway's real closed field $\mathbf{No}$ of surreal numbers is a sweeping
generalization of the real numbers and the ordinals to which a number of
elementary functions such as log and exponentiation have been shown to extend.
The problems of identifying significant classes of functions that can be so
extended and of defining integration for them have proven to be formidable. In
this paper we address this and related unresolved issues by showing that
extensions to $\mathbf{No}$, and thereby integrals, exist for most functions
arising in practical applications. In particular, we show they exist for a
large subclass of the \emph{resurgent functions}, a subclass that contains the
functions that at $\infty$ are semi-algebraic, semi-analytic, analytic,
meromorphic, and Borel summable as well as solutions to nonresonant linear and
nonlinear meromorphic systems of ODEs or of difference equations. By suitable
changes of variables we deal with arbitrarily located singular points. We
further establish a sufficient condition for the theory to carry over to
ordered exponential subfields of $\mathbf{No}$ more generally and illustrate
the result with structures familiar from the surreal literature. The extensions
of functions and integrals that concern us are constructive in nature, which
permits us to work in NBG less the Axiom of Choice (for both sets and proper
classes). Following the completion of the positive portion of the paper, it is
shown that the existence of such constructive extensions and integrals of
substantially more general types of functions (e.g. smooth functions) is
obstructed by considerations from the foundations of mathematics.
