Lectures on integral calculus of functions of one variable and series theory

MINISTRY OF SCIENCE AND HIGHER EDUCATION OF THE RUSSIAN FEDERATION SOUTHERN FEDERAL UNIVERSITY





            Mikhail E. Abramyan


LECTURES ON INTEGRAL CALCULUS OF FUNCTIONS OF ONE VARIABLE AND SERIES THEORY
For students of science and engineering








Rostov-on-Don - Taganrog
Southern Federal University Press
2021

UDC 517.4(075.8)
BBC 22.162я73
  А164

Published by decision of the Educational-Methodical Commission of the I. I. Vorovich Institute of Mathematics, Mechanics, and Computer Science of the Southern Federal University (minutes No. 5 dated April 12, 2021)

Reviewers:
doctor of Physical and Mathematical Sciences, Professor of the Department of Applied Mathematics of the South Russian State Polytechnic University, Honorary official of higher professional education of the Russian Federation,
Professor A. E. Pasenchuk;
candidate of Physical and Mathematical Sciences, Associate Professor of the Department of Algebra and Discrete Mathematics of the I. I. Vorovich Institute of Mathematics, Mechanics, and Computer Science of the Southern Federal University, Docent A. V. Kozak

       Abramyan, M. E.
А164 Lectures on integral calculus of functions of one variable and series theory / M. E. Abramyan ; Southern Federal University. - Rostov-on-Don ; Taganrog : Southern Federal University Press, 2021. - 252 p.
       ISBN 978-5-9275-3829-4

    The textbook contains lecture material for the second part of the course on math-ematical analysis and includes the following topics: indefinite integral, definite inte-gral and its geometric applications, improper integral, numerical series, functional sequences and series, power series, Fourier series. A useful feature of the book is the possibility of studying the course material at the same time as viewing video lectures recorded by the author and available on youtube.com. Sections and subsections of the textbook are provided with information about the lecture number, the start time of the corresponding fragment and the duration of this fragment. In the electronic version of the textbook, this information is presented as hyperlinks, allowing reader to immediately view the required fragment of the lecture.
    The textbook is intended for students specializing in science and engineering.

UDC 517.4(075.8)
ISBN 978-5-9275-3829-4                                   BBC 22.162я73

© Southern Federal University, 2021
                                              © Abramyan M. E., 2021

                Contents





  Preface ............................................................... 7

  Video lectures ........................................................ 9

  1.  Antiderivative and indefinite integral ........................... 13
      Definition of an antiderivative and indefinite integral .......... 13
      Table of indefinite integrals .................................... 14
      The simplest properties of an indefinite integral ................ 15
      Change of variables in an indefinite integral .................... 17
      Formula of integration by parts .................................. 18

  2.  Integration of rational functions ................................ 22
      Partial fraction decomposition of a rational function ............ 22
      Methods for finding the decomposition of a rational function ..... 23
      Integration of terms in the partial fraction decomposition of a rational function ......................................... 24
      Theorem on the integration of a rational function ................ 26

  3.  Integration of trigonometric functions ........................... 28
      Rational expressions for trigonometric functions ................. 28
      Universal trigonometric substitution ............................. 28
      Features of the use of universal trigonometric substitution ...... 29
      Other types of variable change for trigonometric expressions ..... 31

  4.   Integration of irrational functions ............................. 36
      Integration of a rational function with an irrational argument ... 36
      Generalization to the case of several irrational arguments ....... 37
      Integration of the binomial differential ......................... 38
      Euler’s substitutions ............................................ 39

  5.  Definite integral and Darboux sums ............................... 44
      Definite integral ................................................ 44
      Darboux sums and Darboux integrals ............................... 48
      Integrability criterion in terms of Darboux sums ................. 52

M. E. Abramyan. Lectures on integral calculus and series theory

  6.  Classes of integrable functions.
         Properties of a definite integral ............................. 57
      Classes of integrable functions .................................. 57
      Integral properties associated with integrands ................... 61
      Properties associated with integration segments .................. 64
      Estimates for integrals .......................................... 67
      Mean value theorems for definite integrals ....................... 72

  7.  Integral with a variable upper limit.
         Newton-Leibniz formula..........................................76
      Integral with a variable upper limit ............................. 76
      Newton-Leibniz formula ............................................80
      Additional techniques for calculating definite integrals ......... 82

  8.  Calculation of areas and volumes ................................. 87
      Quadrable figures on a plane ..................................... 87
      Area of a curvilinear trapezoid and area of a curvilinear sector . 90
      Volume calculation ............................................... 98

  9.  Curves and calculating their length ............................. 106
      Vector functions and their properties ............................ 106
      Differentiable vector functions .................................. 108
      Lagrange’s theorem for vector functions ......................... 110
      Curves in three-dimensional space. Rectifiable curves ........... 112
      Properties of continuously differentiable curves ................ 114
      Versions of the formula for finding the length of a curve ....... 118

  10.   Improper integrals: definition and properties ................. 121
      Tasks leading to the notion of an improper integral ............. 121
      Definitions of an improper integral ............................. 122
      Properties of improper integrals ................................ 124

  11.   Absolute and conditional convergence of improper integrals 128
      Cauchy criterion for the convergence of an improper integral .... 128
      Absolute convergence of improper integrals ...................... 129
      Properties of improper integrals of non-negative functions ...... 130
      Conditional convergence of improper integrals ................... 134
      Dirichlet’s test for conditional convergence of an improper integral . . 136
      Integrals with several singularities ............................ 138

Contents

5

12.  Numerical series .............................................. 141
   Numerical series: definition and examples ....................... 141
   Cauchy criterion for the convergence of a numerical series and a necessary condition for its convergence .............. 143
   Absolutely convergent numerical series and arithmetic properties of convergent numerical series ... 144

13.  Convergence tests for numerical series with non-negative terms .......................................... 147
   Comparison test ................................................. 147
   Integral test of convergence .................................... 149
   D’Alembert’s test and Cauchy’s test for convergence of a numerical series ...................... 151

14.  Alternating series and conditional convergence ................ 156
   Alternating series .............................................. 156
   Dirichlet’s test and Abel’s test for conditional convergence of a numerical series .......... 159
   Additional remarks on absolutely and conditionally convergent series ........................ 164

15.  Functional sequences and series ............................... 165
   Pointwise and uniform convergence of a functional sequence and a functional series ........... 165
   Cauchy criterion for the uniform convergence of a functional sequence and a functional series ........... 170
   Tests of uniform convergence of functional series ............... 172

16.  Properties of uniformly converging sequences and series . . 176
   Continuity of the uniform limit ................................. 176
   Integration of functional sequences and series .................. 179
   Differentiation of functional sequences and series .............. 182

17.  Power series .................................................. 186
   Power series: definition and Abel’s theorems on its convergence . 186
   Limit inferior and limit superior of a sequence ................. 189
   Cauchy-Hadamard formula for the radius of convergence of a power series ........................... 191
   Properties of power series ...................................... 194

M. E. Abramyan. Lectures on integral calculus and series theory

18.  Taylor series ...................................................... 198
   Real analytic functions and their expansions into Taylor series ...... 198
   Real analytic functions and the property of infinite differentiability . 200 Sufficient condition for the existence of a Taylor series.
      Expansions of exponent, sine, and cosine into a Taylor series ..... 203
   Taylor series expansion of a power function .......................... 206
   Taylor series expansions of the logarithm and arcsine ................ 208
19.  Fourier series in Euclidean space .................................. 212
   Real Euclidean space and its properties .............................. 212
   Fourier series with respect to an orthonormal sequence of vectors in Euclidean space ............................... 216
   Fourier series over a complete orthonormal sequence of vectors ....... 219
20.  Fourier series in the space of integrable functions ................ 224
   Euclidean space of integrable functions .............................. 224
   Constructing an orthonormal sequence of integrable functions ......... 226
   Constructing a formal Fourier series for integrable functions ........ 229
   Convergence of the Fourier series in mean square
      in the case of periodic continuous functions ...................... 231
   Convergence of the Fourier series in mean square
      in the case of piecewise continuous functions ..................... 233
   Pointwise convergence of the Fourier series .......................... 236
   Uniform convergence of the Fourier series ............................ 237
   Decreasing rate of Fourier coefficients for differentiable functions . 242
References .............................................................. 244
Index ................................................................... 246

                                            In memory of Professor Vladimir Stavrovich Pilidi (1946-2021)




            Preface


   The book is a continuation of the textbook [1] and contains lecture material of the second part of the course on mathematical analysis, which was read by the author for several years at the I. I. Vorovich Institute of Mathematics, Mechanics, and Computer Science of the Southern Federal University (specialization 01.03.02 - “Applied Mathematics and Computer Science”). The book includes the following topics: indefinite integral, definite integral and its geometric applications, improper integral, numerical series, functional sequences and series, power series, Fourier series.
   Beyond the scope of the course material presented in [1] and this book, there are topics related to the differential and integral calculus of functions of many variables.
   This book, like the book [1], can be attributed to the category of “short textbooks”, covering only the material that can usually be given in lectures. In this respect, it is similar to books [10, 16] and differs from the “detailed textbooks” that cover the sub ject with much greater completeness. In particular, topics related to the integral calculus of functions of one variable are described in detail in textbooks [4, 6, 8, 11, 14, 18, 19], and topics related to series theory are included in textbooks [4, 6, 7, 9, 12, 14, 18-20]; moreover, the theory of Fourier series is often presented separately (see [5, 13, 15]).
   Most of the statements in the book are provided with detailed proof; for a few auxiliary facts taken without proof, references are given to textbooks in which these facts are proved (the textbook [18] was chosen as the main source for such references).
   Like the book [1], the proposed book has two main features: relationship with the set of video lectures and the presence of two versions: in Russian and English (the Russian version of the book [1] is [2]). The noted features and the additional advantages for the reader resulting from them are described in detail in the preface to [1]. Books [3, 10, 17] can be mentioned as additional sources in English that are closest to Russian textbooks.

M. E. Abramyan. Lectures on integral calculus and series theory

   The index to the book is composed on the same principles as the index to [1]: it contains all definitions and theorems; all references to theorems include their detailed descriptions grouped in the section “Theorem”. In addition, all theorems and other concepts containing surnames in their titles are given in the positions corresponding to these surnames. In the electronic version of the book, page numbers in the index, as well as in the table of contents, are hyperlinks allowing to go directly to this page.
   The initial “Video Lectures” section provides complete information about the set of video lectures related to the book, including their Internet links. This information allows the reader to quickly access the required lecture even in the absence of an electronic version of the book.

            Video lectures


   If the framed text follows the title of the section or subsection, this means that a fragment of the video lecture is associated with this section or subsection. The framed text consists of three parts: the number of the video lecture, the time from which this fragment begins, and the duration of this fragment.
   For example, the following text 2.1A/00:00 (16:47) is located after the title of the first section of Chapter 1 (the section is devoted to the definition of the antiderivative and indefinite integral). It means that this topic is discussed at the very beginning of lecture 2.1A, and the corresponding fragment of the lecture lasts 16 minutes 47 seconds. The last section of Chapter 20 is the section devoted to the decreasing rate of Fourier coefficients for differentiable functions. The correspondent text is 3.19B/33:49 (06:32) , which means that this topic is discussed in the lecture 3.19B, starting at 33:49, and the discussion lasts 6 minutes 32 seconds.
   The double numbering of video lectures is due to the fact that they are taken from two sets with numbers 2 and 3 corresponding to lectures of the second and third semester; the lectures in each set are numbered starting from 1.
   In the electronic version of the book, all framed texts are hyperlinks. Clicking on such text allows you to immediately play the corresponding lecture, starting from the specified time.
   When using the paper version of the book, hyperlinks, of course, are not available, therefore, an additional information is provided here, which will allow you to quickly start playing the required video lecture.
   All video lectures are available on youtube.com. The first 10 video lectures belong to set 2 and are the initial lectures of this set (with numbers from 1 to 10); the final 11 video lectures belong to the middle part of set 3 and have numbers from 9 to 19 in this set. In addition, there is a link to video lecture 2.11A, in which the topic “Curves” ends, and a link to video lecture 3.8B, in which the topic “Improper integrals” begins. All other video lectures consist of two parts: A and B. The following list of lectures contains their titles and short links to each part.

M. E. Abramyan. Lectures on integral calculus and series theory

    2.1. Indefinite integral
        2.1  A: https://youtu.be/66lAeLxskVA
        2.1B  : https://youtu.be/xzIopk1WCDM
    2.2. Integration of rational functions
        2.2  A: https://youtu.be/aLuD104G8PI
        2.2B  : https://youtu.be/pPDP0Lv23fk
    2.3. Integration of trigonometric and irrational functions
        2.3  A: https://youtu.be/_5Maq2J0eHg
        2.3B  : https://youtu.be/aSDoNpfUbAs
    2.4. Definite integral. Darboux sums
        2.4  A: https://youtu.be/TRBKy1OknMM
        2.4B  : https://youtu.be/a4gf4Temgug
    2.5. Classes of integrable functions
        2.5  A: https://youtu.be/oLRSzkV4FLo
        2.5B  : https://youtu.be/OXUliFTV26s
    2.6. Properties of a definite integral
        2.6  A: https://youtu.be/VkS-AcA9njQ
        2.6B  : https://youtu.be/tygGvPGHTps
    2.7. Newton-Leibniz formula
        2.7  A: https://youtu.be/h77yheGoE1I
        2.7B  : https://youtu.be/FPhuVOZFZZ8
    2.8. Calculation of areas
        2.8  A: https://youtu.be/Yg2rrKjorF8
        2.8B  : https://youtu.be/sX5r7CP2oR0
    2.9. Calculation of volumes
        2.9  A: https://youtu.be/3Vpk5JvFLaM
        2.9B  : https://youtu.be/6VT320AFKbw
    2.10. Vector functions. Calculation of a curve length
        2.10  A: https://youtu.be/Q6sxEiXVzhc
        2.10B  : https://youtu.be/xb8oN2tz4Lw
    2.11. Metric spaces
        2.11A: https://youtu.be/J29z4Sog7WE
    3.8. Definition and properties of an improper integral
        3.8B:  https://youtu.be/3r3u9nmPvQI
    3.9. Absolute and conditional convergence of improper integrals
        3.9  A: https://youtu.be/at_eysCbc_M

Video lectures

11

       3.9B  : https://youtu.be/dVh4k6yr8O8
   3.10. Definition and properties of a numerical series, convergence tests
       3.10 A: https://youtu.be/RuNzgI_hUCk
       3.10B  : https://youtu.be/PcIYNHo15_Y
   3.11. Convergence tests (continuation), alternating series
       3.11 A: https://youtu.be/ielvgfjqFjM
       3.11B  : https://youtu.be/l1j-OAwBM5w
   3.12. Functional sequences and series, uniform convergence
       3.12 A: https://youtu.be/vlcY9UpBHGg
       3.12B  : https://youtu.be/PRXEFme2sV0
   3.13. Properties of functional sequences and series
       3.13 A: https://youtu.be/pJywld91FOs
       3.13B  : https://youtu.be/cyHCvVqlDGw
   3.14. Power series
       3.14  A: https://youtu.be/lkbV5-3O7Ps
       3.14B  : https://youtu.be/uOH9-hFgtbM
   3.15. Properties of power series
       3.15 A: https://youtu.be/gvKZiJVjOhE
       3.15B  : https://youtu.be/JzgBm_z7OqI
   3.16. Taylor series
       3.16 A: https://youtu.be/jM7_Gc7vThE
       3.16B  : https://youtu.be/8Js_Dl29pX0
   3.17. Fourier series in Euclidean space
       3.17  A: https://youtu.be/yT2KwZh8XVQ
       3.17B  : https://youtu.be/vnHwF6qCLRU
   3.18. Fourier series in the space of integrable functions
       3.18 A: https://youtu.be/2_hb1tefg7U
       3.18B  : https://youtu.be/yJqsGKaYgmw
   3.19. Properties of Fourier series for various classes of functions
       3.19 A: https://youtu.be/Y6ftB0rijqk
       3.19B  : https://youtu.be/5UJfMwBOpx4
   You can create a link that immediately plays the video lecture, starting from the specified time. Let us describe this additional feature using the previously mentioned fragment 3.19B/33:49 (06:32) as an example. This is a fragment of part B of video lecture 3.19B, its short link has the form 5UJfMwBOpx4. We need to play a lecture starting at 33:49.

M. E. Abramyan. Lectures on integral calculus and series theory

   To do this, use the Internet link https://www.youtube.com/watch? specifying two options after it: a short link to the video lecture (option v=) and the start time of playback (option t=). The options themselves must be separated by the & character.
   In our case, the full text of the Internet link will be as follows:
       https://www.youtube.com/watch?v=5UJfMwBOpx4&t=33m49s
   Pay attention to the time format: after the number of minutes, the letter m is indicated; after the number of seconds, the letter s is indicated. If the number of seconds is 0, then only the number of minutes can be specified.
   A set of hyperlinks to video lectures, which also contains the names of the corresponding chapters, sections, and subsections of this book, is presented on the website mmcs.sfedu.ru of the Institute of Mathematics, Mechanics, and Computer Science of the Southern Federal University (Moodle environment, link http://edu.mmcs.sfedu.ru/course/view.php?id=271 for the lecture set 2 and link http://edu.mmcs.sfedu.ru/course/view.php?id=379 for the lecture set 3). At the top of each specified page, a set of hyperlinks is displayed with titles in Russian and then in English.
   Information related to subtitles for lectures is given in [1] (subsection “Using subtitles” of section “Video lectures”).

                1. Antiderivative and indefinite integral





            Definition of an antiderivative
            and indefinite integral                             2.ia/00:00 (16:47)


   Definition.
   Let the function f be defined on the interval (a, b), where a and b are finite points or points at infinity. Let the function F be a differentiable function on this interval, with F0(x) = f (x) for x 6 (a, b). Then the function F is called the antiderivative (or primitive function) of the function f on a given interval.
   The process of finding an antiderivative is called indefinite integration (or antidifferentiation). If a function has an antiderivative on (a, b), then it is called integrable on (a, b).
   Hereinafter we, as a rule, will not specify interval on which the function is integrable.
   The question arises: how many different antiderivatives exist? Let F1 be the antiderivative of the function f, that is, F₁⁰(x) = f (x). Let F2 (x) = = F1(x) + C , where C is a constant. Then the function F2 is also the antiderivative of the function f , since
       F2(x) = (F1(x) + C)' = Fi(x) = f (x).
   Therefore, if we add a constant to some antiderivative, then we will also get a primitive function. So, there exists an infinite number of antiderivatives, differing from each other by a constant term.
   There are no other antiderivatives: all possible antiderivatives can be obtained by adding a constant to some selected antiderivative. Let us formalize this fact as a theorem.
   Theorem (on antiderivatives of a given function).
   Let F1 and F2 be antiderivatives of f on (a, b). Then there exists a constant C 6 R such that F2(x) = F1(x) + C.
   Proof.
   We introduce the auxiliary function h(x) = F₂(x) - F₁(x). The function h(x) is differentiable on (a, b) as the difference of differentiable functions. Let us find its derivative:

M. E. Abramyan. Lectures on integral calculus and series theory

        h⁰(x) = (F>(x) - Fi(x))⁰ = F2(x) — F0(x) = f (x) - f (x) = 0.

   Thus, h⁰(x) is equal to 0 at any point x € (a,b). Then, by corollary 1 of Lagrange’s theorem [1, Ch. 21], the function h(x) is a constant on the interval (a, b):

        h(x) = C,   x € (a, b).
   Therefore, F2(x) - Fi(x) = C, F2(x) = Fi (x) + C.
   So, knowing one antiderivative, we can obtain all the other antiderivatives, since they all differ from the chosen antiderivative by a constant term.
   Definition.
   The indefinite integral f (x) dx of the function f is the set of all its antiderivatives: if Fi is some antiderivative of the function f (that is, Fᵢ⁰(x) = f (x)), then


        f(x)dx =def {Fi(x) +C, C € R}.


   The symbol is called the integral sign, the function f(x) is called the integrand, and the expression f(x) dx under the integral sign is called the element of integration.
   As a rule, curly braces are not used and, moreover, it is not indicated that C is an arbitrary real constant:


          f(x) dx = Fᵢ (x) + C.



            Table of indefinite integrals



2.1A/16:47 (12:44)

        0 dx = C.

        A dx = Ax + C, A € R.

      /xa+1
        xa dx =—-j—j- + C, x> 0, a € R \{-1}.

      d — dx = In |x| + C, x = 0.
  To prove the last formula, it suffices to differentiate the superposition ln |x| = ln y ◦ |x| for x 6= 0:


(ln |x|)⁰ = (ln y)⁰|y₌|ₓ|

• (|x|)0 =1
y y=|x|

sign x   1
• Sign x = ——— = — |x|                x