| |
| |
Foreword | |
| |
| |
Preface | |
| |
| |
Biographies | |
| |
| |
Introduction | |
| |
| |
Acknowledgment | |
| |
| |
| |
Antiderivative(s) [or Indefinite Integral(s)] | |
| |
| |
| |
Introduction | |
| |
| |
| |
Useful Symbols, Terms, and Phrases Frequently Needed | |
| |
| |
| |
Table(s) of Derivatives and their corresponding Integrals | |
| |
| |
| |
Integration of Certain Combinations of Functions | |
| |
| |
| |
Comparison Between the Operations of Differentiation and Integration | |
| |
| |
| |
Integration Using Trigonometric Identities | |
| |
| |
| |
Introduction | |
| |
| |
| |
Some Important Integrals Involving sin x and cos x | |
| |
| |
| |
Integrals of the Form (dx/(a sin x + b cos x)), where a, b r | |
| |
| |
| |
Integration by Substitution: Change of Variable of Integration | |
| |
| |
| |
Introduction | |
| |
| |
| |
Generalized Power Rule | |
| |
| |
| |
Theorem | |
| |
| |
| |
To Evaluate Integrals of the Form , where a, b, c, and d are constant | |
| |
| |
| |
Further Integration by Substitution: Additional Standard Integrals | |
| |
| |
| |
Introduction | |
| |
| |
| |
Special Cases of Integrals and Proof for Standard Integrals | |
| |
| |
| |
Some New Integrals | |
| |
| |
| |
Four More Standard Integrals | |
| |
| |
| |
Integration by Parts | |
| |
| |
| |
Introduction | |
| |
| |
| |
Obtaining the Rule for Integration by Parts | |
| |
| |
| |
Helpful Pictures Connecting Inverse Trigonometric Functions with Ordinary Trigonometric Functions | |
| |
| |
| |
Rule for Proper Choice of First Function | |
| |
| |
| |
Further Integration by Parts: Where the Given Integral Reappears on Right-Hand Side | |
| |
| |
| |
Introduction | |
| |
| |
| |
An Important Result: A Corollary to Integration by Parts | |
| |
| |
| |
Application of the Corollary to Integration by Parts to Integrals that cannot be Solved Otherwise | |
| |
| |
| |
Simpler Method(s) for Evaluating Standard Integrals | |
| |
| |
| |
To Evaluate | |
| |
| |
| |
Preparation for the Definite Integral: The Concept of Area | |
| |
| |
| |
Introduction | |
| |
| |
| |
Preparation for the Definite Integral | |
| |
| |
| |
The Definite Integral as an Area | |
| |
| |
| |
Definition of Area in Terms of the Definite Integral | |
| |
| |
| |
Riemann Sums and the Analytical Definition of the Definite Integral | |
| |
| |
| |
The Fundamental Theorems of Calculus | |
| |
| |
| |
Introduction | |
| |
| |
| |
Definite Integrals | |
| |
| |
| |
The Area of Function A(x) | |
| |
| |
| |
Statement and Proof of the Second Fundamental Theorem of Calculus | |
| |
| |
| |
Differentiating a Definite Integral with Respect to a Variable Upper Limit | |
| |
| |
| |
The Integral Function Identified as lnx or log<sub>e</sub>x | |
| |
| |
| |
Introduction | |
| |
| |
| |
Definition of Natural Logarithmic Function | |
| |
| |
| |
The Calculus of lnx | |
| |
| |
| |
The Graph of the Natural Logarithmic Function lnx | |
| |
| |
| |
The Natural Exponential Function [exp(x) or e<sup>x</sup>] | |
| |
| |
| |
Methods for Evaluating Definite Integrals | |
| |
| |
| |
Introduction | |
| |
| |
| |
The Rule for Evaluating Definite Integrals | |
| |
| |
| |
Some Rules (Theorems) for Evaluation of Definite Integrals | |
| |
| |
| |
Method of Integration by Parts in Definite Integrals | |
| |
| |
| |
Some Important Properties of Definite Integrals | |
| |
| |
| |
Introduction | |
| |
| |
| |
Some Important Properties of Definite Integrals | |
| |
| |
| |
Proof of Property (P<sub>0</sub>) | |
| |
| |
| |
Proof of Property (P<sub>5</sub>) | |
| |
| |
| |
Definite Integrals: Types of Functions | |
| |
| |
| |
Applying the Definite Integral to Compute the Area of a Plane Figure | |
| |
| |
| |
Introduction | |
| |
| |
| |
Computing the Area of a Plane Region | |
| |
| |
| |
Constructing the Rough Sketch [Cartesian Curves] | |
| |
| |
| |
Computing the Area of a Circle (Developing Simpler Techniques) | |
| |
| |
| |
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 | |
| |
| |
| |
Introduction | |
| |
| |
| |
Methods of Integration | |
| |
| |
| |
Equation for the Length of a Curve in Polar Coordinates | |
| |
| |
| |
Solids of Revolution | |
| |
| |
| |
Formula for the Volume of a "Solid of Revolution" | |
| |
| |
| |
Area(s) of Surface(s) of Revolution | |
| |
| |
| |
Differential Equations: Related Concepts and Terminology | |
| |
| |
| |
Introduction | |
| |
| |
| |
Important Formal Applications of Differentials (dy and dx) | |
| |
| |
| |
Independent Arbitrary Constants (or Essential Arbitrary Constants) | |
| |
| |
| |
Definition: Integral Curve | |
| |
| |
| |
Formation of a Differential Equation from a Given Relation, Involving Variables and the Essential Arbitrary Constants (or Parameters) | |
| |
| |
| |
General Procedure for Eliminating "Two" Independent Arbitrary Constants (Using the Concept of Determinant) | |
| |
| |
| |
The Simplest Type of Differential Equations | |
| |
| |
| |
Methods of Solving Ordinary Differential Equations of the First Order and of the First Degree | |
| |
| |
| |
Introduction | |
| |
| |
| |
Methods of Solving Differential Equations | |
| |
| |
| |
Linear Differential Equations | |
| |
| |
| |
Type III: Exact Differential Equations | |
| |
| |
| |
Applications of Differential Equations | |
| |
| |
INDEX | |