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Impossibility theorems for elementary integration

Liouville proved that certain integrals, most famously ∫ e−x 2 dx, cannot be expressed in elementary terms. We explain how to give precise meaning to the notion of integration “in elementary terms”, and we formulate Liouville’s theorem that characterizes the possible form of elementary antiderivatives. Using this theorem, we deduce a practical criterion for proving such impossibility results in special cases. This criterion is illustrated in two examples: the Gaussian integral ∫ e−x 2 dx that arises in probability theory and the logarithmic integral ∫ dt/ log(t) that arises in the study of primes. Our exposition is aimed at students who are familiar with calculus and elementary abstract algebra (at the level of polynomial rings F [t] over a field F ).algebra (at the level of polynomial rings F [t] over a field F ).