Table of Contents
Preface vii
1 Indefinite and Definite Integrals 1
1.1 Antiderivatives and Indefinite Integral 1
1.1.1 Definitions and Examples 1
1.1.2 Validation of Indefinite Integrals 4
1.1.3 Which Functions Are Integrable? 5
1.1.4 Properties of Indefinite Integral (Integration Rules) 5
1.2 Definite Integral 6
1.2.1 Definitions 6
1.2.2 Which Functions Are Integrable? 9
1.2.3 Properties of Definite Integral (Integration Rules) 9
1.2.4 Integration by Definition 10
1.2.5 Integral Mean Value Theorem 11
1.2.6 Fundamental Theorem of Calculus 12
1.2.7 Total Change Theorem 15
1.2.8 Integrals of Even and Odd Functions 16
2 Direct Integration 17
2.1 Table Integrals and Useful Integration Formula 17
2.2 What Is Direct Integration and How Does It Work? 20
2.2.1 By Integration Rules Only 21
2.2.2 Multiplication/Division Before Integration 21
2.2.3 Applying Minor Adjustments 22
2.2.4 Using Identities 23
2.2.5 Transforming Products into Sums 26
2.2.6 Using Conjugate Radical Expressions 27
2.2.7 Square Completion 28
2.3 Direct Integration for Definite Integral 29
2.4 Applications 31
2.5 Practice Problems 33
3 Method of Substitution 35
3.1 Substitution for Indefinite Integral 35
3.1.1 What for? Why? How? 35
3.1.2 Perfect Substitution 36
3.1.3 Introducing a Missing Constant 37
3.1.4 Trivial Substitution 39
3.1.5 More Than a Missing Constant 40
3.1.6 More Than One Way 42
3.1.7 More Than One Substitution 43
3.2 Substitution for Definite Integral 45
3.2.1 What for? Why? How? 45
3.3 Applications 48
3.4 Practice Problems 49
4 Method of Integration by Parts 51
4.1 Partial Integration for Indefinite Integral 51
4.1.1 What for? Why? How? 51
4.1.2 Three Special Types of Integrals 52
4.1.3 Beyond Three Special Types 55
4.1.4 Reduction Formulas 57
4.2 Partial Integration for Definite Integral 59
4.2.1 What for? Why? How? 59
4.3 Combining Substitution and Partial Integration 62
4.4 Applications 63
4.5 Practice Problems 64
5 Trigonometric Integrals 65
5.1 Direct Integration 65
5.2 Using Integration Methods 66
5.2.1 Integration via Reduction Formulas 66
5.2.2 Integrals of the Form ∫ sinm x cosn x dx 70
5.2.3 Integrals of the Form ∫ tanm x secn x dx 76
5.3 Applications 79
5.4 Practice Problems 81
6 Trigonometric Substitutions 83
6.1 Reverse Substitutions 83
6.2 Integrals Containing a2 - x2 84
6.3 Integrals Containing x2 + a2 88
6.4 Integrals Containing x2 - a2 92
6.5 Applications 96
6.6 Practice Problems 98
7 Integration of Rational Functions 99
7.1 Rational Functions 99
7.2 Partial Fractions 100
7.2.1 Integration of Type 1/Type 2 Partial Fractions 101
7.2.2 Integration of Type 3 Partial Fractions 101
7.2.3 Integration of Type 4 Partial Fractions 103
7.3 Partial Fraction Decomposition 104
7.4 Partial Fraction Method 107
7.5 Applications 112
7.6 Practice Problems 114
8 Rationalizing Substitutions 115
8.1 Integrals with Radicals 115
8.1.1 Integrals of the Form ∫ R (x, $$$ ) dx 115
8.1.2 Integrals of the Form ∫ R (x, xm1/n1, . . ., xmk/nk 116
8.2 Integrals with Exponentials 117
8.3 Trigonometric Integrals 118
8.3.1 Integrals of the Form ∫ R(tan x) dx 118
8.3.2 Integrals of the Form ∫ R(sin x, cos x) dx 119
8.4 Applications 121
8.5 Practice Problems 124
Can We Integrate Them All Now? 125
9 Improper Integrals 127
9.1 Type 1 Improper Integrals (Unbounded Interval) 127
9.1.1 Right-Sided Unboundedness 127
9.1.2 Left-Sided Unboundedness 130
9.1.3 Two-Sided Unboundedness 132
9.2 Type 2 Improper Integrals (Unbounded Integrand) 134
9.2.1 Unboundedness at the Left Endpoint 135
9.2.2 Unboundedness at the Right Endpoint 137
9.2.3 Unboundedness Inside the Interval 138
9.3 Applications 141
9.4 Practice Problems 142
Mixed Integration Problems 145
Answer Key 147
Appendix A Table of Basic Integrals 153
Appendix B Reduction Formulas 155
Appendix C Basic Identities of Algebra and Trigonometry 157
Bibliography 161
Index 163
