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The closed-form integration of arbitrary functions

Consider the usual experience of a student who progresses far enough in school mathematics to begin to study the calculus. After motivation of the topic of “rates of change”, simple differentiation of polynomials is learned. More advanced functions are then considered, and eventually the student meets the product, quotient and chain rules. The result: with enough algebraic accuracy and persistence, the student can determine derived functions for virtually any sufficiently wellbehaved combination of standard functions.