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Foreword | p. ix |
Preface | p. xiii |
Biographies | p. xxi |
Introduction | p. xxiii |
Acknowledgment | p. xxv |
Antiderivative(s) [or Indefinite Integral(s)] | p. 1 |
Introduction | p. 1 |
Useful Symbols, Terms, and Phrases Frequently Needed | p. 6 |
Table(s) of Derivatives and their corresponding Integrals | p. 7 |
Integration of Certain Combinations of Functions | p. 10 |
Comparison Between the Operations of Differentiation and Integration | p. 15 |
Integration Using Trigonometric Identities | p. 17 |
Introduction | p. 17 |
Some Important Integrals Involving sin x and cos x | p. 34 |
Integrals of the Form (dx/(a sin x + b cos x)), where a, b r | p. 37 |
Integration by Substitution: Change of Variable of Integration | p. 43 |
Introduction | p. 43 |
Generalized Power Rule | p. 43 |
Theorem | p. 46 |
To Evaluate Integrals of the Form , where a, b, c, and d are constant | p. 60 |
Further Integration by Substitution: Additional Standard Integrals | p. 67 |
Introduction | p. 67 |
Special Cases of Integrals and Proof for Standard Integrals | p. 68 |
Some New Integrals | p. 84 |
Four More Standard Integrals | p. 85 |
Integration by Parts | p. 97 |
Introduction | p. 97 |
Obtaining the Rule for Integration by Parts | p. 98 |
Helpful Pictures Connecting Inverse Trigonometric Functions with Ordinary Trigonometric Functions | p. 113 |
Rule for Proper Choice of First Function | p. 115 |
Further Integration by Parts: Where the Given Integral Reappears on Right-Hand Side | p. 117 |
Introduction | p. 117 |
An Important Result: A Corollary to Integration by Parts | p. 120 |
Application of the Corollary to Integration by Parts to Integrals that cannot be Solved Otherwise | p. 124 |
Simpler Method(s) for Evaluating Standard Integrals | p. 126 |
To Evaluate | p. 133 |
Preparation for the Definite Integral: The Concept of Area | p. 139 |
Introduction | p. 139 |
Preparation for the Definite Integral | p. 140 |
The Definite Integral as an Area | p. 143 |
Definition of Area in Terms of the Definite Integral | p. 151 |
Riemann Sums and the Analytical Definition of the Definite Integral | p. 151 |
The Fundamental Theorems of Calculus | p. 165 |
Introduction | p. 165 |
Definite Integrals | p. 165 |
The Area of Function A(x) | p. 167 |
Statement and Proof of the Second Fundamental Theorem of Calculus | p. 171 |
Differentiating a Definite Integral with Respect to a Variable Upper Limit | p. 172 |
The Integral Function Identified as lnx or log_{e}x | p. 183 |
Introduction | p. 183 |
Definition of Natural Logarithmic Function | p. 186 |
The Calculus of lnx | p. 187 |
The Graph of the Natural Logarithmic Function lnx | p. 194 |
The Natural Exponential Function [exp(x) or e^{x}] | p. 196 |
Methods for Evaluating Definite Integrals | p. 197 |
Introduction | p. 197 |
The Rule for Evaluating Definite Integrals | p. 198 |
Some Rules (Theorems) for Evaluation of Definite Integrals | p. 200 |
Method of Integration by Parts in Definite Integrals | p. 209 |
Some Important Properties of Definite Integrals | p. 213 |
Introduction | p. 213 |
Some Important Properties of Definite Integrals | p. 213 |
Proof of Property (P_{0}) | p. 214 |
Proof of Property (P_{5}) | p. 228 |
Definite Integrals: Types of Functions | p. 232 |
Applying the Definite Integral to Compute the Area of a Plane Figure | p. 249 |
Introduction | p. 249 |
Computing the Area of a Plane Region | p. 252 |
Constructing the Rough Sketch [Cartesian Curves] | p. 257 |
Computing the Area of a Circle (Developing Simpler Techniques) | p. 272 |
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 | p. 295 |
Introduction | p. 295 |
Methods of Integration | p. 295 |
Equation for the Length of a Curve in Polar Coordinates | p. 300 |
Solids of Revolution | p. 302 |
Formula for the Volume of a "Solid of Revolution" | p. 303 |
Area(s) of Surface(s) of Revolution | p. 314 |
Differential Equations: Related Concepts and Terminology | p. 321 |
Introduction | p. 321 |
Important Formal Applications of Differentials (dy and dx) | p. 323 |
Independent Arbitrary Constants (or Essential Arbitrary Constants) | p. 331 |
Definition: Integral Curve | p. 332 |
Formation of a Differential Equation from a Given Relation, Involving Variables and the Essential Arbitrary Constants (or Parameters) | p. 333 |
General Procedure for Eliminating "Two" Independent Arbitrary Constants (Using the Concept of Determinant) | p. 338 |
The Simplest Type of Differential Equations | p. 357 |
Methods of Solving Ordinary Differential Equations of the First Order and of the First Degree | p. 361 |
Introduction | p. 361 |
Methods of Solving Differential Equations | p. 362 |
Linear Differential Equations | p. 388 |
Type III: Exact Differential Equations | p. 397 |
Applications of Differential Equations | p. 398 |
INDEX | p. 399 |
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