Table of Contents

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

About the Author

ULRICH L. ROHDE, PhD, ScD, Dr-Ing, is Chairman of SynergyMicrowave Corporation, President of Communications ConsultingCorporation, and a Partner of Rohde & Schwarz. A Fellow of theIEEE, Professor Rohde holds several patents and has published morethan 200 scientific papers. G. C. JAIN, B.Sc., is a retired scientist from theDefense Research and Development Organization in India. AJAY K. PODDAR, PhD, is Chief Scientist at SynergyMicrowave Corporation. A Senior Member of the IEEE, Dr. Poddarholds several dozen patents and has published more than 180scientific papers. A. K. GHOSH, PhD, is Professor in the Department ofAerospace Engineering at the IIT Kanpur, India. He has publishedmore than 120 scientific papers.

Reviews

Introduction to Integral Calculus is an excellent bookfor upper-undergraduate calculus courses and is also an idealreference for students and professionals who would like to gain afurther understanding of the use of calculus to solve problems in asimplified manner. (Zentralblatt MATH,2012) Long on examples but often short of exercises, this workmight best be used as a reference source. Summing Up:Recommended. Lower-and upper-divisionundergraduates. (Choice, 1 September 2012)