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 $\text {1896}$

Subject Matter

• Calculus

• Differential Equations

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 $\ds \int \frac {\d x} {\sqrt R}$

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

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