A Treatise on the Integral Calculus with Applications, Examples and Problems [I, Reprint 1954 ed.]
136 25
English Pages XXII;922 [945] Year 1954
Title Page
Preface
Table of Contents
Abbreviations used in the References.
CHAPTER I - NATURE OF THE PROBLEM. PRELIMINARY CONSIDERATIONS.
CHAPTER II - STANDARD FORMS.
CHAPTER III - CHANGE OF THE INDEPENDENT VARIABLE.
CHAPTER IV - INTEGRATION BY PARTS. POWERS OF SINES AND COSINES.
CHAPTER V - RATIONAL ALGEBRAIC FRACTIONAL FORMS.
CHAPTER VI - INTEGRALS OF FORMS: Int(1/(a + b*cos(x) + c*sin(x))^n, dx)
CHAPTER VII - FURTHER REDUCTION FORMULAE.
CHAPTER VIII - FORM Int( F(x, Sqrt(R), dx), WHERE R IS QUADRATIC.
CHAPTER IX - GENERAL THEOREMS.
CHAPTER X - DIFFERENTIATION OF A DEFINITE INTEGRALWITH REGARD TO A PARAMETER.
CHAPTER XI - PRELIMINARY TO INTEGRATION OF Int(M/(N Sqrt(Q)), dx) WHERE Q IS A QUARTIC. DEFINITIONS OF ELLIPTIC FUNCTIONS.
CHAPTER XII - QUADRATURE (I).PLANE SURFACES. CARTESIANS AND POLARS.
CHAPTER XIII - QUADRATURE (II).
CHAPTER XIV - QUADRATURE (III).
CHAPTER XV - QUADRATURE (IV). MISCELLANEOUS THEOREMS.
CHAPTER XVI - RECTIFICATION (I).
CHAPTER XVII - RECTIFICATION (II). APPLICATION OF ELLIPTIC FUNCTIONS.
CHAPTER XVIII - RECTIFICATION (III). MISCELLANOUS THEOREMS.
CHAPTER XIX - MOVING CURVES.
CHAPTER XX - RECTIFCATION OF TWISTED CURVES.
CHAPTER XXI - VOLUMES OF REVOLUTION, ETC.
CHAPTER XXII - SURFACES AND VOLUMES IN GENERAL.
Answers to Examples and Problems.
Chelsea Scientific Books 1954
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CONTENTS. CHAPTER I. NATURE OF THE PROBLEM. PRELIMINARY CONSIDERATIONS. PA.GBS
ARTS.
1-8. 9-15. 16. 17-19. 20. 21.
22-23. 24-25. 26.
Fundamental Notions. Fluents and Fluxions, Problem to be attacked Newton's Second and Third Lemmas. Analytical Expression. Notation illustrative Examples The Fundamental Proposition Unknown Curve through Specified Points Simpson's Rule Trapezoidal Rule, Weddle's Rule, etc. Volumes of Revolution Mechanical Integration. General Review -
1-3
4-7 8-12 13-17 17-19 19-20 21-22 22-25 26-28 28-39
PROBLEMS
CHAPTER II. STANDARD FORMS. 40
36-38. 39-42. 43-45.
Reversal of Differentiation Nomenclature. Constant of Integration. Inverse Notation Laws satisfied by ])-1. Integration of Series. Geometrical Illustrations Integration of x n, X-I, (ax+b)n, (ax+b)-1 Forms cp(x)/(ax+b), cp'(x)/cp(x), (cpx)ncp'(x), F'(cpx)cp'(x)
49-50
TABLE OF RESULTS
52-53
46.
GENERAL REMARKS
54-56
27-28. 29-32. 33-35.
41-42 43-47 47-48
56-66
PROBLEMS
ix
x
CONTENTS.
CHAPTER III. CHANGE OF THE INDEPENDENT VARIABLE. PAGE
ARTS.
47·51. 52-54. 55-58. 59-68. 69. 70-73. 74·76.
Mode of Effecting a Change of Variable. Alteration of the Limits Case of a Multiple. Valued Function Purpose and Choice of a Substitution. The Hyperbolic Functions, Direct and Inverse. Properties The Gudermannian and its Inverse . As to Tables of the Inverse Gudermannian, the Hyperbolic Functions, etc. Integration of cosec x, sec x, cosech x, seoh x, (a cos x+b sin X)-l
67·69 69-71 71-74 76-84 84-85 85-87 88-89
Integration of (all _xll)f'!, (xll+all)f'l, (Xl _all)H, sec3x, cosec3x
89-91
80·84.
J~~,
91-94
85.
J JR
86·87.
dx =2sinh-l-Y~, JJx(adx-x) =2ain-l-Y~,a JJx(a+x) a
77-79.
(R:=azll + 2bx+ c) ; various forms; IJRdx
A X+ B
94
dx
JJ-x(xdx-a) =2cosh-l-Y~a and other forms
88. 89.
.
Visible Relation between the Integrand and the Integral ADDITIONAL LIST OF STANDARD RESULTS PROBLEMS
95 95-96 96-97 99-104
CHAPTER IV INTEGRATION BY PARTS. 90-93. 94-96. 97. 99. 100. 101.
102.
POWERS OF SINES AND COSINES.
Integration by Parts. The Method and Rule Rule for Repeated Operation of Integration by Parts Forma ea:l: sin bz sin ex sin da, ea:l: sin" x coss z, ea:l: sin" x cos nx, etc. . Integration of an Inverse Function Geometrical Consideration of Integration by Parts General Idea of a Reduction Formula Integration of x m sin nx, xm cos nx
105...107 108-109 110 111 111-112 113 113-114
xi
CONTENTS. AB'!8.
103. 104-105. 106-11I. 112-1I3. 1I4-126.
Integration of xne ax sin bx, xne ax COB bx Integration of eax cos" bx, eax sin" bx Integration of x'7'(log x)n A Trigonometrical Process. Multiple Angles Powers and Products of Powers of Sines and Cosines, with or without an Exponential Factor PROBLEMS
PAGBB
115 1I5-117 117-119 119-121 121-131 131-137
CHAPTER V. RATIONAL ALGEBRAIC FRACTIONAL FORMS. 1 1 1 1 127-129. Forms al_ XS ' xl-al' a2+x2' ,82+(x+a)2' 1 1 ,as - (x+a)2' (x+a)2 - ,B2
138-139
130-135. Integration ftjf. (R::=ax2+bx+c); various cases and
136-138.
f
forms q PX.t dx •
139-141 141-143 NOTE ON PARTIAL FRACTIONS.
139-141. General Statement of the Case 142-143. Partial Fraction corresponding to an Unrepeated Linear Factor 144-146. Linear Factors Repeated The Coefficients expressed as Repeated Differentiations 147. 148-149. Conditions that 150. 151. 152-154. 155-156. 157-159.
f~~:~ dx may be purely Algebraic
Irreducible Quadratic Factors Unrepeated Irreducible Quadratic Factors Repeated General Typical Form of the Result and its Integration Use of Indeterminate Coefficients Modifications for Special Cases -
160-165. Cases of
fIT
n x dx
143-144 145-146 146-148 148 149-150 150 151 152-153 153-154 154-156 156-158
-
(x 2+ar ll )
1
166-167. 168-169.
f
xmdx
tc
x1n _ 2anxn COB na+aln' e .
f:1~dx, PROBLEMS
where
~=(a+,8xY'
-
158-159 160-161 161-169
xii
CONTENTS.
CHAPTER VI. INTEGRALS OF FORM !(a+bcosC:+csinX)ft' etc. ARTS.
f
j ----,--------,---
dx dx - -dx --, . a+bcosx' a+bsinx a+b cos z-l-c sin x , dx ' dx 180-181. Forms Ja+bcoshx' Ja+bsinhx'
170·179. Forms
J
PAGE
170·176
176-178
Ja+bcoshtt;+csinhx 182-184. Integration of the above Forms expressed in Terms of the Integrand
179-18]
185-187. Reduction Formulae for J(a+b~os x)ft' J(a+t:in x)ft'
J(a+bcosC:+CSinX)ft 188-189. Corresponding Reduction Formulae for Hyperbolic Functions 190-193. Integration of Fractions of Forms a+b cos 6 +c sin 0 a+b cos O+c sin 0 a 1 +b 1 cos (J+c1 sin O· (a 1 +b1 cos O+c1 sin (J)ft'
182-185 185-186