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1.
基于集合理论中的同余类概念,给出不定积分的新定义,解决了在原定义下存在的不定积分式间运算意义不明确等问题,使不定积分的定义变得准确合理.  相似文献   

2.
本文对2018年全国硕士研究生入学统一考试中一道不定积分问题进行了研究,给出了5种解法,提出了一种新的计算不定积分的方法——待定函数法.  相似文献   

3.
将传统不定积分表中的递推公式全部展开,并将近200个常用不定积分公式合并为43个积分公式,从而给出一张新不定积分表.  相似文献   

4.
积分学是微积分的重要组成部分,而不定积分是整个积分学的基础,是运算的核心部分.本文主要通过历史和逻辑的角度对不定积分符号进行分析,并给出注记.  相似文献   

5.
在微积分中,不定积分的计算既是重点又是难点.巧妙地利用1=sin~2x+cos~2x介绍几类三角函数的不定积分,讨论形如Lmn=∫sec mxcscnxdx,m,n=0,1,2,…的不定积分.同时,讨论其它相关三角函数的不定积分,如∫sec~mxdx,∫csc~mxdx,∫secx csc~mxdx,∫sec~mxcscxdx,∫tan~mxdx,∫cot ~mxdx,m=0,12,….另外,通过几个实例简单讨论以上积分公式的应用.  相似文献   

6.
通过对几道不定积分例题细致分析,阐明凑法可简化特殊类型函数的不定积分的解题过程,达到事半功倍的效果,也有利于提高学生灵活解题的能力和激发他们的学习兴趣.  相似文献   

7.
本文从简单的三角函数有理式不定积分出发,通过合理添加参数的方法,得出含参数的三角函数有理式的不定积分计算.  相似文献   

8.
在求不定积分时,当被积函数中含有某已知函数的反函数时,若令反函数为积分变量,作换元积分法,往往使不定积分更容易求得.下面举例说明.  相似文献   

9.
<正> 单元函数积分学包含不定积分和定积分这两部分内容,其中原函数、不定积分和定积分的概念是其基本概念,积分中值定理、上限是变量的定积分及其求导定理是其基本理论,而Newton-Lebniz公式是其基本公式,积分法是其基本的运算法.本文将侧重围绕着积分学的基本概念和基本理论,论述三个关系,即原函数与不定积分;不定积分与定积分;原函数的存在性与可积性的关系以及积分中值定理.  相似文献   

10.
本文给出了函数的商的不定积分公式,揭示了该公式的应用方法.  相似文献   

11.
基于积分运算是微分运算的逆运算,利用线性代数方法,提出了一种求解几类特殊函数不定积分的新方法,并结合实例验证了方法的有效性.  相似文献   

12.
A substantial number of new indefinite integrals involving the incomplete elliptic integral of the third kind are presented, together with a few integrals for the other two kinds of incomplete elliptic integral. These have been derived using a Lagrangian method which is based on the differential equations which these functions satisfy. Techniques for obtaining new integrals are discussed, together with transformations of the governing differential equations. Integrals involving products combining elliptic integrals of different kinds are also presented.  相似文献   

13.
A substantial number of indefinite integrals are presented for the incomplete elliptic integrals of the first and second kinds. The number of new results presented is about three times the total number to be found in the current literature. These integrals were obtained with a Lagrangian method based on the differential equations which these functions obey. All results have been checked numerically with Mathematica. Similar results for the incomplete elliptic integral of the third kind will be presented separately.  相似文献   

14.
针对求解三角函数有理式不定积分的问题,借助实例说明传统方法存在不足之处,并给出此类不定积分的完整解.  相似文献   

15.
《Quaestiones Mathematicae》2013,36(4):365-380
Abstract

For an arbitrary infinite-dimensional Banach space 𝔛, we construct examples of strongly-measurable 𝔛-valued Pettis integrable functions whose indefinite Pettis integrals are nowhere weakly differentiable; thus, for these functions the Lebesgue Differentiation Theorem fails rather spectacularly. We also relate the degree of nondifferentia-bility of the indefinite Pettis integral to the cotype of 𝔛, from which it follows that our examples are reasonably sharp.  相似文献   

16.
This study was designed to micro-analyze the role of the design of interactive diagrams in learning-teaching the concept of the indefinite integral. The presented case study focuses on the engagement of one pair of 17-year-old students with an interactive diagram for graphically learning-teaching the indefinite integral concept. The authors performed three rounds of analysis to detect the mathematical elements involved in finding connections between the function graphs and the antiderivative function graphs and to identify how the design features of the interactive diagram functioned in the students’ engagement processes. The data analysis identified how the pedagogical functions designed in the interactive calculus diagram were reflected in the students’ learning-teaching processes with the diagram.  相似文献   

17.
The Henstock-Kurzweil and McShane product integrals generalize the notion of the Riemann product integral. We study properties of the corresponding indefinite integrals (i.e. product integrals considered as functions of the upper bound of integration). It is shown that the indefinite McShane product integral of a matrix-valued function A is absolutely continuous. As a consequence we obtain that the McShane product integral of A over [a, b] exists and is invertible if and only if A is Bochner integrable on [a, b]. Supported by grant No. 201/04/0690 of the Grant Agency of the Czech Republic.  相似文献   

18.
This paper reports on the use of APOS theory to investigate conceptual understanding of the indefinite integral amongst undergraduate students at the University of KwaZulu-Natal, South Africa. We present a Preliminary Genetic Decomposition (PGD) for the indefinite integral, which predicts the mental constructions and mechanisms that may facilitate conceptual understanding. In this pilot study, the analysis of students’ written responses to the research instrument suggested that more than half of the participants lacked the prerequisite knowledge of the concepts of function, derivative of functions and the chain rule. The findings confirmed that students experience greater difficulty dealing with transcendental functions than with algebraic functions. The analysis indicated that the cognitive mechanism of reversal, to recognise the inverse nature of differentiation and integration, was inconsistent or absent in many students. Interview data from a subset of participants was employed for triangulation with the document analysis. The empirical data suggested refinement of the PGD and modification of the research instrument, and further fine-grained interviews with study participants to investigate their conceptual understanding more deeply, and for the purposes of data triangulation.  相似文献   

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