HENSTOCK- KURZWEIL INTEGRAL ON [a,b]

The theory of the Riemann integral was not fully satisfactory. Many important functions do not have a Riemann integral. So, Henstock and Kurzweil make the new theory of integral. From the background, the writer will be research about Henstock-Kurzweil integral and also theorems of HenstockKurzweil Integral. HenstockKurzweil Integral is generalized from Riemann integral. In this case the writer uses research methods literature or literature study carried out by way explore, observe, examine and identify the existing knowledge in the literature. In this thesis explain about partition which used in HenstockKurzweil Integral, definition and some property of HenstockKurzweil Integral. And some properties of HenstockKurzweil integral as follows: value of the HenstockKurzweil integral is unique, linearity of the Henstock-Kurzweil integral, Additivity of the Henstock-Kurzweil integral, Cauchy criteria, nonnegativity of Henstock-Kurzweil integral and primitive function.


INTRODUCTION
We have already mentioned the developments, during the 1630's, by Fermat and Descrates leading to analytic geometry and the theory of the derivatives. However, the subject we know as calculus did not begin to take shape until the late 1660's when Issac Newton (1642-1727) created his theory of fluxions and invented the method of inverse tangents to find areas under curves. The reversal of the process for finding tangent lines to find areas was also discovered in the 1680's by Leibniz (1646-1716), who was unaware of Newton unpublished work and who arrived at the discovery by a very different route. Leibniz introduced the terminology calculus differential and calculus integral, since finding tangents lines involved differences and finding areas involved summations. Thus they had discovered that integration, being a process of summation, was inverse to the operation of differentiation.
During a century and a half of development and refinement of techniques, calculus consisted of these paired operations and their applications, primarily to physical problems. In the 1850s, Bernhard Riemann (1826-1866) adopted a new and different viewpoint. He separated the concept of integration from its companion, differentiation, and examined the motivating summation and limit process of finding areas by itself. He broadened the scope by considering all functions on an interval for which this process of integration could be defined: the class of integrable functions. The fundamental Theorem of calculus became a result that held only for a restricted set of integrable functions. The viewpoint of Riemann led others to invent other integration theories, the most significant being Lebesgue's theory of integration.
The theory of the Riemann integral was not fully satisfactory. Many important functions do not have a Riemann integral even after we extend the class of integrable functions slightly by allowing "improper" Riemann integrals. For example Characteristic function.
In 1957, the Czech mathematician Jaroslav Kurzweil discovered a new definition of this integral elegantly similar in nature to Riemann's original definition which he named the gauge integral; the theory was developed by Ralph Henstock. Due to these two important mathematicians, it is now commonly known as Henstock-Kurzweil integral. The simplicity of Kurzweil's definition made some educators advocate that this integral should replace the Riemann integral in introductory calculus courses, but this idea has not gained traction.
Concerning the background of the study, the writer formulates the statement of the problems as follows: 1

Supremum and Infimum
We start with a straightforward definition similar to many others in this course. Read the definitions carefully, and note the use of ≤ and ≥ here rather than < and > .

Definition 1
Let S be a subset of ℜ Let S be a subset of ℜ .
1. If S is bounded above, then an upper bound u is said to be supremum (or a least upper bound) of S if no number smaller than u is an upper bound of S. 2. If S is bounded below, then a lower bound w is said to be infimum (or a greatest lower bound) of S if no number greater than w is a lower bound of S.

Limit of Function
The essence of the concept of limit for real valued functions of a real variable is this: if L is a real number, then can be made as close to L as we wish by taking x sufficiently close to x0. This is made precise in the following definition.

Definition 3
We say that f(x) approaches the limit L as x approaches 0 x , and write

Continuity
Definition 5 a) We say that f is continuous at 0 if, in addition.

Definition 7
for all but finitely many points 0 If c) fails to hold at some 0

Uniform Continuity
If the a Lipschitz conditions satisfied with constant K, then given Therefore f is uniformly continuous on A.

Upper And Lower Integral
Definition 10 Is the infimum of all upper sums. The lower sum of f over P is Is the supremum off all lower sums.

Riemann Integral
Riemann integral, defined in 1854, was the first of the modern theories of integration and enjoys many of the desirable properties of an integration theory.
The mesh of the partition is then the length of the largest Riemann began by considering the approximating (Riemann) sums , : is Riemann integrable This definition defines the integral as a limit of sums as the mesh of the partition approaches 0.
Since ε was arbitrary, it follows that Thus, the value of the integral is unique.
Note that any δ-fine division is also δ1-fine and δ2-fine.
Therefore for any δ-fine division The proof is complete.
for each i .
Where the first sum is over D and the second over D'. Proof ( ) ⇒ we will prove that if A function is Henstock- Analogous to the situation for real-valued sequences, the condition that      The Riemann sum of f over And similarly that over Since ε is arbitrary, we have the required inequality.

Theorem 11
If f is Henstock-Kurzweil integrable on [ ] Then the difference is an error. The definition of the Henstock-Kurzweil integral says that the absolute error is also small, whereas Henstock'Lemma. In fact, the two are equivalent by theorem 3.3.8. Another way of putting it is that taking any partial sum 1 Σ of Σ we still have ( ) ( ) ( )( ) That is to say, the selected error is again small, and indeed it is equivalent to the above two.
Consequently the result follows.