  In this paper, we study the consequences of the fundamental theorem of
calculus from an algebraic point of view. For functions with singularities,
this leads to a generalized notion of evaluation. We investigate properties of
such integro-differential rings and discuss many examples. We also construct
corresponding integro-differential operators and provide normal forms via
rewrite rules. They are then used to derive several identities and properties
in a purely algebraic way, generalizing well-known results from analysis. In
identities like shuffle relations for nested integrals and the Taylor formula,
additional terms are obtained that take singularities into account. Another
focus lies on treating basics of linear ODEs in this framework of
integro-differential operators. These operators can have matrix coefficients,
which allow to treat systems of arbitrary size in a unified way. In the
appendix, using tensor reduction systems, we give the technical details of
normal forms and prove them for operators including other functionals besides
evaluation.
