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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, the antiderivatives of certain elementary functions cannot themselves be expressed as elementary functions.It should be realized that the notion of an elementary function is merely a matter of convention. One could choose to add the error function to the list of elementary functions, and with this new list, the antiderivative of e x2 is elementary.The machinery of differential Galois theory allows one to determine when an elementary function does or does not have an antiderivative that can be expressed as an elementary function. Differential Galois theory is a theory based on the model of Galois theory. Whereas algebraic Galois theory studies extensions of algebraic fields, differential Galois theory studies extensions of differential fields, i.e. fields that are equipped with a derivation, D.