Lectures on Integral Calculus of Functions of One Variable and Series Theory (378,00 руб.)

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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 Мathematics 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 А164 Abramyan, M. E. 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. ISBN 978-5-9275-3829-4 UDC 517.4(075.8) BBC 22.162я73 © Southern Federal University, 2021 © Abramyan M. E., 2021

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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

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4 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

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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

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6 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

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