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62. Various examples of integration of exponential functions
63, 64. Integration of logarithmic functions
65
66
SECTION 2.-Circular Functions.
65-67. Integration of fundamental circular functions..
68, 69. Integration of (sin x)" dx and (cos x)" dx..
70. Integration of (sin x)-"dx and of (cosx)-"dx
71, 72. Integration of (sinx)" (cosx)" dx
73. Integration of (tanx)"dx and of (cotx)"dx
74. Integration of x" sin x dx, and of x cos x dx
75, 76. Integration of eax (cos x)" dx, and of eax (sinx)" dx
77. Integration of f(x) sin-1xdx, f(x) tan-1x dx, &c.
79. Integration by substitution
..
80. Reason why the number of indefinite integrals is so small
CHAP. IV.
ON DEFINITE INTEGRATION AND ON DEFINITE INTEGRALS.
SECTION 1.-Definite Integrals determined by Indefinite Integration.
SECTION 2.-The Change of Limits in Definite Integrals.
83. A general definite integral expressed in its complete form
84. The effect of a reversal of the limits
91
92
85. The value of a definite integral is unaltered, if the extreme
limits are the same and the intermediate limits are con-
tinuously additive
86. Examples in illustration and application of the theorem
87. An extension of the theorem given in Art. 85
88. Theorems as to mean values of limits
89, 90. Consideration of the case wherein the element-function
takes an infinite value for a value of the variable be-
tween the limits ..
91. Cauchy's principal value of a definite integral
92. Apparent anomalies removed by the correct theory of definite
integration
98
101
102
SECTION 3.-The Transformation of a Definite Integral by a
95. Simplification of definite integrals by means of transformation 107
SECTION 4.—The Differentiation and Integration of a Definite
Integral with respect to a Variable Parameter.
96. Variation of a definite integral due to the variation of a
constant contained in the limits and in the element-
function..
108
97. Interchange of this process with that of the definite integra-
tion without alteration of value in the integral ..
98. Evaluation of certain definite integrals by this process..
99. Integral of a definite integral with respect to a parameter
contained in the element-function
100. Evaluation of certain definite integrals by the process
101. Further application of the preceding processes ..
111
113
115
SECTION 5.-Definite Integrals involving Impossible Quantities.
Cauchy's Method of Evaluation.
102. The value of a definite integral when the element-function
becomes infinite for a value of its variable within the
105. The correction for infinity and discontinuity
106. Application of the theorem to particular forms of element-
functions
107. Cases where the correction for infinity vanishes
127
130
133
135
138
108. The values of the preceding when the limits are changed..
109. Another form of function treated on Cauchy's method
110. Remarks on the method
SECTION 6.-Methods of Approximating to the value of a Definite
Integral.
111. Evaluation by direct summation
112. Evaluation by summation of terms at finite intervals
113. Geometrical interpretation of the process..
114. Application of the process to Mensuration
115. Approximation by means of known integrals
116. Approximate values given by the forms of definite integrals 147
117. Bernoulli's series for approximate value ..
118. Approximate value deduced from Taylor's series.
119. The theory of integration by series..
120, 121. Maclaurin's series applied to integration
SECTION 7.-The Gamma-Function, and other Allied Integrals.
122. Definition of the gamma-function; various forms of it
123. The gamma-function is determinate and continuous
124. Particular values of the gamma-function for particular
values of the argument
125. Definition of the beta-function, and its relation to the
127. The proof of the theorem г(n + 1) = nг (n)
166
167
128. Another proof of the same for positive and integral values
129. The proof of the theorem г (n) г(1—n) =
131. Determination of the x-differential of log г (x) ..
132. The third fundamental theorem of the gamma-function
133. Euler's constant; Gauss' definition of the gamma-function 170
134. The numerical calculation of г (n)
136. The deduction of the fundamental theorems of the gamma-
function from Gauss' definition of the function
173
137. г(n+1)=(2) "+e", when ʼn = ∞,
138, 139. Evaluation of certain definite integrals by means of
the gamma-function
140, 141. The evaluation of other definite integrals by means of
174
176
179
181
183
185
186
142, 143. Application of the gamma-function to the summation
of series in terms of a definite integral
145. The Gaussian series, and various cases of it
146. The value of the gamma-function, when the argument is
negative
SECTION 8.-The Logarithm-Integral.
147. Definition of the logarithm-integral, and its expansion
148. Various applications of the logarithm-integral.
151-153. The calculus of operations applied to successive inte-
gration
192
CHAP. VI.
THE APPLICATION OF SINGLE INTEGRATION TO QUESTIONS OF
GEOMETRY.
SECTION 1.-Rectification of Plane Curves; Rectangular Coordinates.
154. Investigation of the general expression of the length-
156. Examples of rectification by the aid of subsidiary angles.. 202
157, 158. The lengths of elliptic arcs
SECTION 2.-Rectification of Plane Curves; Polar Coordinates.
161. Investigation of the general expression of the length-
element
209
162. Examples of rectification ..
163. Application of length-element in terms of and p
SECTION 3.-The Rectification of Curves in Space.
164. Determination of the length-element; rectangular coor-
165. Determination of the length-element; polar coordinates..
SECTION 4.-Properties of Curves depending on the Length of the arc: the Intrinsic Equation of a curve.
166. Determination of the equation of a curve when 8 is a given
function of x and y
167, 168. The intrinsic equation of a curve
210
211
212
214
217
219
SECTION 5.-Involutes of Plane Curves.
169, 170. Investigation of general properties of involutes.
222
171. Examples of involutes
224
172. Involutes of curves referred to polar coordinates, and ex-
amples
226
SECTION 6.-Geometrical Problems solved by means of Single
THE APPLICATION OF SINGLE DEFINITE INTEGRATION TO THE
THEORY OF SERIES.
SECTION 1.-The Convergence and Divergence of Series.
177. The necessity of a further inquiry into the theory of series
178. The general forms and symbols of series. The definitions
of convergence and divergence of series
179. On series in geometrical and harmonical progression
180. If the terms of a series decrease in magnitude and are
alternately positive and negative, the series is con-
vergent
235
236
237
240