Integration For Calculus, Analysis, And Differential Equations. Techniques, Examples, and Exercises

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Contents Preface 1. Indefinite and Definite Integrals 1.1. Antiderivatives and Indefinite Integral 1.1.1. Definitions and Examples 1.1.2. Validation of Indefinite Integrals 1.1.3. Which Functions Are Integrable? 1.1.4. Properties of Indefinite Integral (Integration Rules) 1.2. Definite Integral 1.2.1. Definitions 1.2.2. Which Functions Are Integrable? 1.2.3. Properties of Definite Integral (Integration Rules) 1.2.4. Integration by Definition 1.2.5. Integral Mean Value Theorem 1.2.6. Fundamental Theorem of Calculus 1.2.7. Total Change Theorem 1.2.8. Integrals of Even and Odd Functions 2. Direct Integration 2.1. Table Integrals and Useful Integration Formula 2.2. What Is Direct Integration and How Does It Work? 2.2.1. By Integration Rules Only 2.2.2. Multiplication/Division Before Integration 2.2.3. Applying Minor Adjustments 2.2.4. Using Identities 2.2.5. Transforming Products into Sums 2.2.6. Using Conjugate Radical Expressions 2.2.7. Square Completion 2.3. Direct Integration for Definite Integral 2.4. Applications 2.5. Practice Problems 3. Method of Substitution 3.1. Substitution for Indefinite Integral 3.1.1. What for? Why? How? 3.1.2. Perfect Substitution 3.1.3. Introducing a Missing Constant 3.1.4. Trivial Substitution 3.1.5. More Than a Missing Constant 3.1.6. More Than One Way 3.1.7. More Than One Substitution 3.2. Substitution for Definite Integral 3.2.1. What for? Why? How? 3.3. Applications 3.4. Practice Problems 4. Method of Integration by Parts 4.1. Partial Integration for Indefinite Integral 4.1.1. What for? Why? How? 4.1.2. Three Special Types of Integrals 4.1.3. Beyond Three Special Types 4.1.4. Reduction Formulas 4.2. Partial Integration for Definite Integral 4.2.1. What for? Why? How? 4.3. Combining Substitution and Partial Integration 4.4. Applications 4.5. Practice Problems 5. Trigonometric Integrals 5.1. Direct Integration 5.2. Using Integration Methods 5.2.1. Integration via Reduction Formulas 5.2.2. Integrals of the Form sinm x cosn x dx 5.2.3. Integrals of the Form tanm x secn x dx 5.3. Applications 5.4. Practice Problems 6. Trigonometric Substitutions 6.1. Reverse Substitutions 6.2. Integrals Containing a2 x2 6.3. Integrals Containing x2 + a2 6.4. Integrals Containing x2 a2 6.5. Applications 6.6. Practice Problems 7. Integration of Rational Functions 7.1. Rational Functions 7.2. Partial Fractions 7.2.1. Integration of Type 1/Type 2 Partial Fractions 7.2.2. Integration of Type 3 Partial Fractions 7.2.3. Integration of Type 4 Partial Fractions 7.3. Partial Fraction Decomposition 7.4. Partial Fraction Method 7.5. Applications 7.6. Practice Problems 8. Rationalizing Substitutions 8.1. Integrals with Radicals 8.1.1. Integrals of the Form dx 8.1.2. Integrals of the Form R (x, xm1/n1, . . . , xmk/nk) dx 8.2. Integrals with Exponentials 8.3. Trigonometric Integrals 8.3.1. Integrals of the Form R(tan x) dx 8.3.2. Integrals of the Form R(sin x, cos x) dx 8.4. Applications 8.5. Practice Problems 9. Improper Integrals 9.1. Type 1 Improper Integrals (Unbounded Interval) 9.1.1. Right-Sided Unboundedness 9.1.2. Left-Sided Unboundedness 9.1.3. Two-Sided Unboundedness 9.2. Type 2 Improper Integrals (Unbounded Integrand) 9.2.1. Unboundedness at the Left Endpoint 9.2.2. Unboundedness at the Right Endpoint 9.2.3. Unboundedness Inside the Interval 9.3. Applications 9.4. Practice Problems Mixed Integration Problems Answer Key Chapter 2: Direct Integration Chapter 3: Method of Substitution Chapter 4: Method of Integration by Parts Chapter 6: Trigonometric Substitutions Chapter 7: Integration of Rational Functions Chapter 8: Rationalizing Substitutions Chapter 9: Improper Integrals Mixed Integration Problems Appendix A Table of Basic Integrals Appendix B Reduction Formulas Appendix C Basic Identities of Algebra and Trigonometry Bibliography Index