scholarly journals VI. On the multiplication of definite integrals

1875 ◽  
Vol 23 (156-163) ◽  
pp. 120-121
Keyword(s):  
Definite Integral ◽  
Definite Integrals ◽  
Straight Lines

The definite integral ∫ y 1 y 0 ∫ x 1 x 0 P dx dy may be considered geometrically as the integral ∫ P dx dy extended over an area bounded by the straight lines whose equations are x = y 1 , x = y 0 , y = x 1 , y = x 0 . Now conceive the axes transformed through an angle of 45°, so that x = ξ/√2 = ƞ/√2, y = ξ/√2 + ƞ/√2; then the equations to the four straight lines become ξ/√2 - ƞ/√2 = y 1 , ξ/√2 - ƞ/√2 = y 0 , ξ/√2 + ƞ/√2 = x 1 , ξ/√2 + ƞ/√2 = x 0 .

scholarly journals Generalizations and applications of Young’s integral inequality by higher order derivatives

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Jun-Qing Wang ◽  
Bai-Ni Guo ◽  
Feng Qi
Keyword(s):  
Differential Geometry ◽  
Integral Inequality ◽  
Higher Order ◽  
Integral Inequalities ◽  
List Type ◽  
Definite Integral ◽  
Exponential Integral ◽  
Definite Integrals ◽  
Logarithmic Integral

Abstract In the paper, the authors generalize Young’s integral inequality via Taylor’s theorems in terms of higher order derivatives and their norms, and apply newly-established integral inequalities to estimate several concrete definite integrals, including a definite integral of a function which plays an indispensable role in differential geometry and has a connection with the Lah numbers in combinatorics, the exponential integral, and the logarithmic integral.

scholarly journals Definite Integral of Algebraic, Exponential and Hyperbolic Functions Expressed in Terms of Special Functions

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1425
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Closed Form ◽  
Special Function ◽  
Special Functions ◽  
Complex Numbers ◽  
Definite Integral ◽  
Hyperbolic Functions ◽  
Closed Form Solutions ◽  
Definite Integrals ◽  
Arbitrary Complex ◽  
Famous Book

While browsing through the famous book of Bierens de Haan, we came across a table with some very interesting integrals. These integrals also appeared in the book of Gradshteyn and Ryzhik. Derivation of these integrals are not listed in the current literature to best of our knowledge. The derivation of such integrals in the book of Gradshteyn and Ryzhik in terms of closed form solutions is pertinent. We evaluate several of these definite integrals of the form ∫0∞(a+y)k−(a−y)keby−1dy, ∫0∞(a+y)k−(a−y)keby+1dy, ∫0∞(a+y)k−(a−y)ksinh(by)dy and ∫0∞(a+y)k+(a−y)kcosh(by)dy in terms of a special function where k, a and b are arbitrary complex numbers.

scholarly journals Definite Integral of Power and Algebraic Functions in terms of the Lerch Function

2021 ◽  
Vol 14 (3) ◽  
pp. 980-988
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Present Note ◽  
Special Functions ◽  
Integral Formula ◽  
Definite Integral ◽  
Algebraic Functions ◽  
Definite Integrals ◽  
Cauchy’S Integral Formula ◽  
Logarithmic Integral

Bierens de haan (1867) evaluated a definite integral involving the cotangent function and this result was also listed in Gradshteyn and Ryzhik (2007). The objective of this present note is to use this integral along with Cauchy's integral formula to derive a definite logarithmic integral in terms of the Lerch function. We will use this integral formula to produce a table of known and new results in terms of special functions and thereby expanding the list of definite integrals in both text books.

scholarly journals On approximation of definite integrals by compound quadrature formulas using derivatives

2021 ◽  
pp. 88-104
Author(s):  
В.В. Шустов
Keyword(s):  
Numerical Methods ◽  
Integration Method ◽  
Quadrature Formulas ◽  
Definite Integral ◽  
Definite Integrals ◽  
Nodal Points ◽  
Point Integration ◽  
The Given ◽  
Maclaurin Formula

Рассмотрена задача вычисления определенного интеграла функции, для которой известны значения ее самой и набора производных до заданного порядка в точках отрезка интегрирования. Построены составные квадратурные формулы, которые используют значения функции и ее производных до m-го порядка включительно. Получено представление остаточного члена, выраженное через производную соответствующего порядка и число узловых точек. Приведены примеры интегрирования заданных функций с исследованием погрешности и ее оценки. Дано сравнение с известными численными методами и формулой Эйлера-Маклорена, которое показало повышенную точность и лучшую сходимость метода двухточечного интегрирования The problem of computing a definite integral of a function for which the values of itself and the set of derivatives up to a given order at the points of the interval of integration are known is considered. Composite quadrature formulas are constructed that use the values of the function and its derivatives up to the m-th order inclusive. A representation of the remainder is obtained, expressed in terms of the derivative of the corresponding order and the number of nodal points. Examples of integration of the given functions with the study of the error and its estimation are given. A comparison is made with the known numerical methods and the Euler-Maclaurin formula, which showed increased accuracy and better convergence of the two-point integration method.

II. On certain definite integrals. No. 8

1881 ◽  
Vol 31 (206-211) ◽  
pp. 330-336
Keyword(s):  
Definite Integral ◽  
Definite Integrals

I commence this paper with some general reflections on the theory of definite integrals. A definite integral may be written thus— ∫ β α dx f ( a, b, c ... x )=Ø( a, b, c ..). If we expand in terms of ( a ) and equate the coefficients of on we shall have ∫ β α dx f 1 ( n, b, c ... x )=Ø( n, b, c ..). * These have been calculated for line A.

scholarly journals On the Comparison of Gauss and Hybrid Methods and their Application to Calculation of Definite Integrals

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Necessary Conditions ◽  
Hybrid Methods ◽  
Definite Integral ◽  
Implicit Methods ◽  
Definite Integrals ◽  
Advantages And Disadvantages ◽  
Chebyshev Method ◽  
Gauss Method ◽  
Gauss Quadrature Formula ◽  
To Receive

As is known there is the wide class of methods for calculation of the definite integrals constructed by the well-known scientists as Newton, Gauss, Chebyshev, Cotes, Simpson, Krylov and etc. It seems that to receive a new result in this area is impossible. The aim of this work is the applied some general form of hybrid methods to computation of definite integral and compares that with the Gauss method. The generalization of the Gauss quadrature formula have been fulfilled in two directions. One of these directions is the using of the implicit methods and the other is the using of the advanced (forward-jumping) methods. Here have compared these methods by shown its advantages and disadvantages in the results of which have recommended to use the implicit method with the special structure. And also are constructed methods, which have applied to calculation of the definite integral with the symmetric bounders. As is known, one of the popular methods for calculation of the definite integrals with the symmetric bounders is the Chebyshev method. Therefore, here have defined some relations between of the above mentioned methods. For the application constructed, here methods are defined the necessary conditions for its convergence. The receive results have illustrated by calculation the values for some model integral using the methods with the degree p  8.

scholarly journals Table in Gradshteyn and Ryzhik: Derivation of definite integrals of a Hyperbolic Function

2021 ◽  
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Special Function ◽  
Special Functions ◽  
Trigonometric Function ◽  
Hyperbolic Function ◽  
Definite Integral ◽  
Definite Integrals ◽  
Catalan’S Constant ◽  
Infinite Sums ◽  
Logarithmic Functions ◽  
Mathematical Constants

We present a method using contour integration to derive definite integrals and their associated infinite sums which can be expressed as a special function. We give a proof of the basic equation and some examples of the method. The advantage of using special functions is their analytic continuation which widens the range of the parameters of the definite integral over which the formula is valid. We give as examples definite integrals of logarithmic functions times a trigonometric function. In various cases these generalizations evaluate to known mathematical constants such as Catalan’s constant and π

The Moi Formula for Improper Exponential Definite Integrals

1994 ◽  
Vol 79 (3) ◽  
pp. 1123-1127 ◽  
Author(s):  
Frank O'Brien ◽  
Sherry E. Hammel ◽  
Chung T. Nguyen
Keyword(s):  
Probability Distribution ◽  
General Formula ◽  
Integral Formula ◽  
Definite Integral ◽  
Definite Integrals ◽  
Finite Moments

A general formula is developed for solving a type of improper exponential definite integral of order n in the number plane. Termed the Moi Formula, it is shown to produce substantially simpler derivations of the finite moments of a probability distribution employed for assessing stochastic randomness, such as recently published by the authors. Other applications of the integral formula are discussed.

scholarly journals A CHAOTIC BAT ALGORITHM FOR SOLVING DEFINITE INTEGRAL

2015 ◽  
Vol 14 (4) ◽  
pp. 5592-5598 ◽  
Author(s):  
Khalil AL-Wagih
Keyword(s):  
Numerical Simulation ◽  
Convergence Rate ◽  
Numerical Integration ◽  
Bat Algorithm ◽  
High Accuracy ◽  
Definite Integral ◽  
Definite Integrals ◽  
High Convergence Rate ◽  
Simulation Results

In this paper, an Improved Bat Algorithm with Chaos (IBACH) is presented for solving definite integral. The IBACH satisfies the question of parallel calculating numerical integration in engineering and those segmentation points that are adaptive. Several numerical simulation results show that the algorithm offers an effective way to calculate numerical value of definite integrals; it has a high convergence rate, a high accuracy and robustness.

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