Fundamental Theorem of Calculus and Computations on Some Special Henstock-Kurzweil Integrals

The constructive definition usually begins with a function f, then by the process of using Riemann sums and limits, we arrive the definition of the integral of f, ∫ b a f. On the other hand, a descriptive definition starts with a primitive F satisfying certain condition(s) such as F ′ = f and F is absolutely continuous if f is Lebesgue integrable, and F is generalized absolutely continuous if f is Henstock-Kurzweil integrable. For descriptive integrals, the deficiency is that we need to know primitive F for which F ′ = f and satisfying some properties. For constructive integration, we proposed in [8] using an uneven partition to get a broader family functions which includes some improper Riemann integrals. In this paper, we describe how we can make use of the Fundamental Theorem of Calculus and the constructive definition to reach a description definition for some improper Riemann integrable functions that are monotonic or highly oscillating with singularity on one end.