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

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

Published $\text {1896}$

### 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

#### 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 $\ds \int \frac {\d x} {\sqrt R}$
Powers and Products of Sines and Cosines
Powers of Secants or Cosecants
Powers of Tangents or Cotangents
Integration of $\ds \int \frac {\d x} {a + b \cos x}$. etc.

#### Chapter VII. Reduction Formulae.

Integration of $x^{m - 1} X^p$, where $X = a + b x^n$
Reduction Formulae for $\ds \int x^{m - 1} X^p \d x$
Reduction Formulae for $\ds \int \sin^p x cos^q x \d x$
Evaluation of $\ds \int_0^{\frac \pi 2} \sin^n x \d x$, $\ds \int_0^{\frac \pi 2} \sin^p x cos^q x \d x$

#### Chapter VIII. Miscellaneous Methods.

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

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