THE INTEGRAL ` A LA HENSTOCK

Abstract

Henstock provided a unified approach to many integrals in use. The author put on record what he knows about the theory and its development. Finally, he gives a personal view on the future of the theory. 1 A Unified Approach Calculus is a gateway to advanced mathematics. The key concepts in calculus are derivative and anti-derivative. The integral is defined as an anti-derivative. It is often called the Newton integral. Hence the chain rule in differentiation becomes integration by substitution in integration, and the derivative of the product of two functions becomes integration by parts in integration. To integrate a function, we differentiate another function so that its derivative is the given function. This other function is called the primitive of the given function. Then the Newton integral of a given function is its primitive. If we integrate then differentiate or differentiate then integrate, we get back to the same function. This property is known as the fundamental theorem of calculus. The integral is regarded as a mapping of a function into another function, namely the primitive. Another approach to integration is taking the generalized limit of Riemann sums. What we have defined is called the Riemann integral. We often define the Riemann integral on a finite interval (a, b). Unfortunately, there are functions that are Newton integrable on (a, b) but not Riemann integrable there. There are also functions that are Riemann integrable and not Newton integrable on (a, b). When we talk about the Riemann integral, we often think of it as a mapping of a function into an integral value. The fundamental theorem of calculus does not hold for the Riemann integral without imposing further conditions. The condition imposed is usually continuity. That is, if we integrate a continuous function into its indefinite integral then when we differentiate we obtain the original continuous function. The next step after the Riemann integral is to introduce the improper Riemann integral. We need the improper integral for applications. The improper Riemann integral includes the Riemann integral, and intersects with the Newton integral. Again the Newton integral and the improper Riemann integral do not include each other.