SECTION 1.-Exponential and Logarithmic Functions.

59. Integration of a dx and emx dx

60, 61. Integration of "ex dx, and x-"ex dx

62. Various examples of integration of exponential functions

63, 64. Integration of logarithmic functions

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 (cosx)" dx, and of eax (sin x)" dx
77. Integration of f(x) sin ̄1xdx, f(x) tan-1x dx, &c.

80

78. Integration of

dx

(a+b cos x)"

80

79. Integration by substitution

81

80. Reason why the number of indefinite integrals is so small

83

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

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

91

92

93

94

..

95

96

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

98

101

102

92. Apparent anomalies removed by the correct theory of definite

integration

SECTION 3.-The Transformation of a Definite Integral by a

change of variable.

93. The theory and its application

94. Examples of transformation ..

103

104

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..

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 ..

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

range ..'

103. Cauchy's method of evaluation, and examples

104. Another case of Cauchy's method

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

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

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

125. Definition of the beta-function, and its relation to the

163

gamma-function...

126. Equivalent forms of the beta-function

127. The proof of the theorem r(n+1) = nr (n)

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
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 π)3 n2+i e−”, when ʼn = ∞,

174

138, 139. Evaluation of certain definite integrals by means of

the gamma-function

176

140, 141. The evaluation of other definite integrals by means of

the gamma-function

179

142, 143. Application of the gamma-function to the summation

of series in terms of a definite integral

181

183

185

144. Similar application of the beta-function

145. The Gaussian series, and various cases of it

146. The value of the gamma-function, when the argument is

SECTION 8.-The Logarithm-Integral.

147. Definition of the logarithm-integral, and its expansion

148. Various applications of the logarithm-integral..

187

189

191

191

151-153. The calculus of operations applied to successive inte-

gration

CHAP. VI.

THE APPLICATION OF SINGLE INTEGRATION TO QUESTIONS OF

192

GEOMETRY.

SECTION 1.-Rectification of Plane Curves; Rectangular Coordinates.

154. Investigation of the general expression of the length-

element

197

155. Examples of rectification ..

198

156. Examples of rectification by the aid of subsidiary angles..

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 ..

210

163. Application of length-element in terms of and p

211

SECTION 3.-The Rectification of Curves in Space.

164. Determination of the length-element; rectangular coor-

dinates

165. Determination of the length-element; polar coordinates..

SECTION 4.-Properties of Curves depending on the Length of the

are the Intrinsic Equation of a curve.

8

166. Determination of the equation of a curve when s is a given
function of x and y..

167, 168. The intrinsic equation of a curve

SECTION 5.-Involutes of Plane Curves.

212

214

217

219

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