TITLE: Abstract algebra of integration theories.
QUESTION [6 upvotes]: Simple group theory is a generalization of basic addition and multiplication of numbers.  In this vein, is there an abstract generalization of integration which does not make reference to the mechanics of the specific integration calculations (e.g. measure theory) but instead characterizes the essence of integration and allows it to be defined in other contexts?

REPLY [3 votes]: Differential Galois theory is one algebraic theory of integration. It tries to explain which differential equations can be solved in elementary means like ordinary Galois theory explains which polynomial equations can be solved be elementary means (that is, by radicals). In particular, it differential Galois theory tries to explain which functions can be integrated in elementary terms.