[eBook Code] Introduction to Integral Calculus (eBook Code, 1st)

FOREWORD ix

PREFACE xiii

BIOGRAPHIES xxi

INTRODUCTION xxiii

ACKNOWLEDGMENT xxv

1 Antiderivative(s) [or Indefinite Integral(s)] 1

1.1 Introduction 1

1.2 Useful Symbols, Terms, and Phrases Frequently Needed 6

1.3 Table(s) of Derivatives and their corresponding Integrals 7

1.4 Integration of Certain Combinations of Functions 10

1.5 Comparison Between the Operations of Differentiation and Integration 15

2 Integration Using Trigonometric Identities 17

2.1 Introduction 17

2.2 Some Important Integrals Involving sin x and cos x 34

2.3 Integrals of the Form ? (d/( a sin  + b cos x)), where a, b 

ϵ r 37

3a Integration by Substitution: Change of Variable of Integration 43

3b Further Integration by Substitution: Additional Standard Integrals 67

4a Integration by Parts 97

4b Further Integration by Parts: Where the Given Integral Reappears on Right-Hand Side 117

5 Preparation for the Definite Integral: The Concept of Area 139

5.1 Introduction 139

5.2 Preparation for the Definite Integral 140

5.3 The Definite Integral as an Area 143

5.4 Definition of Area in Terms of the Definite Integral 151

5.5 Riemann Sums and the Analytical Definition of the Definite Integral 151

6a The Fundamental Theorems of Calculus 165

6b The Integral Function Ð x 1 1 t dt, (x > 0) Identified as ln x or loge x 183

7a Methods for Evaluating Definite Integrals 197

7b Some Important Properties of Definite Integrals 213

8a Applying the Definite Integral to Compute the Area of a Plane Figure 249

8b To Find Length(s) of Arc(s) of Curve(s), the Volume(s) of Solid(s) of Revolution, and the Area(s) of Surface(s) of Solid(s) of Revolution 295

9a Differential Equations: Related Concepts and Terminology 321

9a.4 Definition: Integral Curve 332

9b Methods of Solving Ordinary Differential Equations of the First Order and of the First Degree 361

INDEX 399