# Book:Joseph Edwards/Integral Calculus for Beginners: With an Introduction to the Study of Differential Equations

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## Joseph Edwards:

#### Chapter XIV. Equations of the First Order (

## Joseph Edwards: *Integral Calculus for Beginners: With an Introduction to the Study of Differential Equations*

Published $\text {1896}$.

### Subject Matter

### Contents

#### Preface

### Integral Calculus

#### Chapter I. Notation, Summation, Applications.

- Determination of an Area

- Integration from the Definition

- Volume of Revolution

#### Chapter II. General Method, Standard Forms.

- Fundamental Theorem

- Nomenclature and Notation

- General Laws obeyed by the Integrating Symbol

- Integration of $x^n$, $x^{-1}$

- Table of Results

#### Chapter III. Method of Substitution.

- Method of Changing the Variable

- The Hyperbolic Functions

- Additional Standard Results

#### Chapter IV. Integration by Parts.

- Integration "by Parts" of a Product

- Geometrical Proof

- Extension of the Rule

#### Chapter V. Partial Fractions.

- Standard Cases

- General Fraction with Rational Numerator and Denominator

#### Chapter VI. Sundry Standard Methods.

- Integration of $\displaystyle \int \frac {\d x} {\sqrt R}$

- Powers and Products of Sines and Cosines

- Powers of Secants or Cosecants

- Powers of Tangents or Cotangents

- Integration of $\displaystyle \int \frac {\d x} {a + b \cos x}$. etc.

#### Chapter VII. Reduction Formulae.

- Integration of $x^{m - 1} X^p$, where $X = a + b x^n$

- Reduction Formulae for $\displaystyle \int x^{m - 1} X^p \d x$

- Reduction Formulae for $\displaystyle \int \sin^p x cos^q x \d x$

- Evaluation of $\displaystyle \int_0^{\frac \pi 2} \sin^n x \d x$, $\displaystyle \int_0^{\frac \pi 2} \sin^p x cos^q x \d x$

#### Chapter VIII. Miscellaneous Methods.

- Integration of $\displaystyle \int \frac {\map \phi x \d x} {X \sqrt Y}$

- Integration of some Special Fractional Forms

- General Propositions and Geometrical Illustrations

- Some Elementary Definite Integrals

- Differentiation under an Integral Sign

#### Chapter IX. Rectification.

- Rules for Curve-Tracing

- Formulae for Rectification and Illustrative Examples

- Modification for a Closed Curve

- Arc of an Evolute

- Intrinsic Equation

- Arc of Pedal Curve

#### Chapter X. Quadrature.

- Cartesian Formula

- Sectorial Areas. Polars

- Area of a Closed Curve

- Other Expressions

- Area between a Curve, two Radii of Curvature and the Evolute

- Areas of Pedals

- Corresponding Areas

#### Chapter XI. Surfaces and Volumes of Solids of Revolution.

- Volumes of Revolution

- Surfaces of Revolution

- Theorems of Pappus

- Revolution of a Sectorial Area

#### Chapter XII. Second-order Elements of Area. Miscellaneous Applications.

- Surface Integrals, Cartesian Element

- Centroids; Moment sof Inertia

- Surfaces Integrals, Polar Element

- Centroids, etc., Polar Formulae

### Differential Equations

#### Chapter XIII. Equations of the First Order.

- Genesis of a Differential Equation

- Variables Separable

- Linear Equations

#### Chapter XIV. Equations of the First Order (*Continued*).

- Homogeneous Equations

- One Letter Absent

- Clairaut's Form

#### Chapter XV. Equations of the Second Order. Exact Differential Equations.

- Linear Equations

- One Letter Absent

- General Linear Equation. Removal of a Term

- Exact Differential Equations

#### Chapter XVI. Lear Differential Equation with Constant Coefficients.

- General Form of Solution

- The Complementary Function

- The Particular Integral

- An Equation Reducible to Linear Form with Constant Coefficients

#### Chapter XVII. Orthogonal Trajectories. Miscellaneous Equations.

- Orthogonal Trajectories

- Some Important Dynamical Equations

- Further Illustrative Examples