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

Subject Matter

 * Calculus


 * Differential Equations

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

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