On the K-Riemann integral and Hermite–Hadamard inequalities for K-convex functions

In the present paper we introduce a notion of the K-Riemann integral as a natural generalization of a usual Riemann integral and study its properties. The aim of this paper is to extend the classical Hermite–Hadamard inequalities to the case when the usual Riemann integral is replaced by the K-Riemann integral and the convexity notion is replaced by K-convexity

Kajian Intgeral Henstock

A study of the properties of Henstock integral has been carried out. Henstock integral is an extension of Riemann integral. This is because the Henstock integral is built on the concept of Riemann integral, using the Riemann sum over the partition on the domain interval of a function. The difference lies in controlling the partition. On Riemann integral the control of a partition is roled by a constant positive number, whereas on Henstock integral the control of a partition is roled by possitive function. In this final project, we will learn how to construct the Henstock integral based on the concept of the Riemann integral, the properties of Henstock integral with its example and its relationship with the Riemann integral. The result of the study shows that every function that Riemann integrable is Henstock integrable, but it does not necessarily apply conversely

Vanishing of the integral of the Hurwitz zeta function

A proof is given that the improper Riemann integral of δ(s, a) with respect to the real parameter a, taken over the interval (0, 1], vanishes for all complex s with R(s) < 1. The integral does not exist (as a finite real number) when R(s) ≥ 1