A typical source of mistakes that frequently lead to a wrong or incomplete solution for the antiderivative of a given real function of one real variable is a misuse of the technique of change of variable. The increasing implementation of software in apparently mechanic tasks such as the calculation of antiderivatives has not improved the situation, yet those software packages issue generic warnings such as 'the answer's is not guaranteed to be continuous' or 'the solution might be only valid for parts of the function'. The practical meaning of those vague machine messages is clearly envisaged in this book, which shows how to handle the technique of change of variable in order to provide correct solutions.This book is monographically focused on elementary antidifferentiation and reasonably self-contained, yet it is written in a 'hand-book' style: it has plenty of examples and graphics in an increasing level of difficulty; the most standard changes of variable are studied and the hardest theoretic parts are included in a final Appendix. Each practical chapter has a list of exercises and solutions.This book is intended for instructors and university students of Mathematics of first and second year.
Author(s): Antonio Martinez-Abejón
Publisher: World Scientific
Year: 2020
Language: English
Pages: 282
City: Singapore
Contents
Preface
1. Background
1.1 Notation and basic concepts
1.2 Sequences of real numbers
1.3 Compactness in R
2. Real-valued functions of a real variable
2.1 Limits of functions
2.2 Continuous functions
2.3 Differentiable functions
2.4 Integrable functions
3. Elementary real-valued functions
3.1 Power functions and root functions
3.2 Polynomials
3.3 The exponential function and the log function
3.4 General exponentiation
3.5 The circular functions and their inverse functions
3.6 The hyperbolic functions and their inverse functions
4. Antidifferentiation
4.1 Antiderivatives and primitives
4.2 Fundamental theorems of antidifferentiation. Change of variable
4.3 Important results concerning a change of variable
4.4 A practical method to perform a change of variable
4.5 Continuous functions with no elementary antiderivative
4.6 Basic primitives
5. Antidifferentiation by parts
5.1 Antidifferentiating by parts
5.2 Examples
5.3 Exercises
6. Rational functions
6.1 Fractions with denominator of degree one and numerator of degree zero
6.2 Fractions with irreducible denominator of degree two and numerator of degree zero
6.3 Fractions with irreducible denominator of degree two and numerator of degree one
6.4 Fractions of negative degree whose denominator only has simple real roots
6.5 Fractions of negative degree whose denominator has only simple roots but at least one is not real
6.6 Fractions of negative degree whose denominator has only real roots but at least one is multiple
6.7 Fractions of negative degree whose denominator has at least one non-real root and some multiple root
6.8 Fractions of non-negative degree
6.9 Exercises
7. Fractions of polynomials over? ax2 + bx + c
7.1 Fractions 1/ ax2 + bx + c
7.2 Fractions (Ax + Bq/ ax2 + bx + c
7.3 Fractions P(x)/ ax2 + bx + c for any polynomial P(x)
7.4 Exercises
8. Fractions 1(ax+b)p ax2+bx+c, p N
8.1 The change of variable t = 1/ (ax + )
8.2 Primitives of 1(px)+ px2+bx+c, p N
8.3 Exercises
9. Rational functions of x and of rational powers of ax+bcx+d
9.1 The domain of R(x, (ax+b)(cx+d)p1/q1, . . . , (ax+b)(cx+d) pn/qn)
9.2 The change of variable g(x) = (ax+b)(cx+d) 1/q, 1< q N
9.3 The primitive of R(x, (ax+b)(cx+d) p1/q1 , . . . , (ax+b)(cx+d) pn/qn)
9.4 Exercises
10. Binomial Differentials
10.1 The domain of a binomial differential
10.2 Changes of variable for binomial differentials
10.3 Classification of binomial differentials
10.4 The primitive of a binomial differential of type I
10.5 The primitive of a binomial differential of type II
10.6 Exercises
11. Rational functions of trigonometric arguments
11.1 The domain of R(sin x; cos x)
11.2 Trigonometric changes of variable
11.3 The primitive of R(sin x; cos x)
11.4 The case R(sin x; cos x) R(– sin x, –cos x)
11.5 The case R(– sin x, cos x) = –R(sin x, cos x)
11.6 The case R(sin x, –cos x) = –R(sin x, cos x)
11.7 Exercises
12. Functions R(x, ax2 + bx + c)
12.1 The domain of R(x, ax2+bx+c)
12.2 Changes of variable associated with R(x, ax2+bx+c)
12.3 Functions R(x, 1 – x2)
12.4 Functions R(x, 1 + x2)
12.5 Functions R(x, x2 - 1)
12.6 Exercises
13. Tables
13.1 Elementary methods of antidifferentiation
13.2 Trigonometric identities
Appendix A Complements
A.1 Some observations on the real-valued n-th root functions in connection with the complex numbers
A.2 The role of the fundamental functions in Real Analysis
A.3 Notes on the construction of the functions log x and ex
A.4 Notes on the construction of the circular functions
A.5 Some facts about the relation between differentiability and integrability
A.6 Geometrical representation of complex numbers and the n-th complex roots of unity
A.7 Long division of polynomials
A.8 Synthetic division of polynomials
A.9 Completing squares
A.10 The Fundamental Theorem of Algebra
A.11 Roots of polynomials of degree greater than two
A.12 Decomposition of fractions of polynomials
A.13 Cardinality of sets
Bibliography
Index
