It has been said that integration is an art, whereas differentiation is merely a skill. Even beginning calculus students quickly recognize that integration is a fundamentally much harder problem than differentiation.

In fact, in the 19-th century Liouville proved that certain simple kinds of functions do not have anti-derivatives that can be written down “in elementary terms.” (Of course, one has to give a reasonable and precise definition of what “elementary” means.) This is analogous to the fact that the roots to many polynomials of degree above four can't be written down exactly using standard arithmetic operations and root extractions. There is a powerful analogy between the theories of solving for roots of polynomials and solving certain kinds of differential equations (integrating).

The main new idea that we need to understand for Liouville's theorem is a differential field. Assuming just knowledge of calculus and an interest in understanding how on earth one could really prove that a function doesn't have an “elementary” anti-derivative, we will introduce differential fields and apply them to the solvability of some concrete differential equations. In particular, we will see why e^−x2 cannot be integrated in elementary terms.

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