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

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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


Answers