In integration we have the following dichotomy. We spend most of our time teaching the Riemann integral but we do most of our research with the Lebesgue integral. Students do not see the Lebesgue integral until their Senior year or their first year of graduate studies. This means that only those who study a considerable amount of mathematics will have the benefits of using the Lebesgue theory.

The Henstock integral alleviates this pain. It includes the Riemann, improper Riemann and Lebesgue integrals but is of the same level of difficulty as the Riemann integral. It has the following properties:

• It has an easy definition in terms of Riemann sums, not requiring any measure theory.
• Every derivative is Henstock integrable. This is not true of Riemann or Lebesgue integrals! Thus, we have the most complete version of the Fundamental Theorem of Calculus and Stokes's Theorem.
• The Henstock integral is nonabsolute. For a function to be integrable it is not necessary for its absolute value to be integrable.

Besides being easy to teach, the Henstock integral is a useful research tool. It can be defined in abstract spaces, there are appropriate convergence theorems, integration with respect to Schwartz distributions is natural and we have a normed space structure.

This talk will be an elementary introduction to nonabsolute (Henstock) integration. Besides discussing undergraduate teaching we will bring forth examples from harmonic functions, potential theory, Fourier analysis, differential/difference equations and quantum path integrals of phenomena that, by their very nature, are best described using Henstock integration.