FLUENTS

SECTION I. Definitions.—II. General Theorems.—III. Rational Fluxions.—IV. Irrational Fluxions.—V. Circular Fluxions.—VI. Logarithmic Fluxions.—VII. Exponential Fluxions.—VIII. Index of Fluxions.

THE fluents of such expressions, as are the most likely to occur in the solution of physical problems, may be very conveniently arranged in the form of a TABLE; the principal materials of which will be extracted from Meier Hirsch's Integraltafeln. 4. Berlin, 1810. It might have been somewhat enlarged by additional matter that may be found in the earlier publications of our countrymen Waring and Landen, which have been particularly consulted on the occasion; but Waring's improvements relate most commonly to cases so complicated, as seldom to be applicable to practical purposes; and Landen's theorems, though incomparably more distinct and better arranged than Waring's, tend rather to the investigation of some elegant analogies, than to the facilitation of actual computations. Some of these, however, will be briefly noticed, and an improvement in the mode of notation will be attempted, which, if universally adopted, would tend to save much unnecessary circumlocution in the enunciation of many general theorems.

1. The earlier letters of the alphabet, as far as q, and sometimes r, are commonly employed to denote constant quantities; the subsequent letters generally for quantities considered as variable. They are here employed as relating indifferently to quantities positive or negative, and to numbers whole or fractional; except when they are used as indices or exponents.

2. The Italic character is employed, in preference to others, for denoting quantities in general, the Roman for characteristic marks, as d for a fluxion, or differential, sin, cos, or f, ϕ, for sine and cosine; and

hl for hyperbolic logarithm. The long Italic $\backslash int$, however, not being otherwise used, serves very conveniently as a characteristic, to denote a fluent.

3. When the Italic letters m, n, p, q, r, or any others, are employed as indices, they are to be here understood as denoting any numbers without limitation; the Roman small letters, m, n, will be applied to whole numbers only, excluding fractions, but either positive or negative, or 0; the small Italic Capitals M, N, to positive numbers, whether whole or fractional, excluding negative numbers only; and the small Roman Capitals M, N, to positive integers only, including however 0.

4. The characteristic $\backslash Sigma$ implies the sum of a finite number of terms, derived from all the possible variations of a quantity, which is here denoted by a small letter of the Greek alphabet.

5. A comma, in an index, denotes or.

6. The fluents, indicated by the table, are to be understood as corresponding equally to any particular values of the quantities concerned; so that, in order to obtain the expression of the definite quantity required by the conditions of any problem, we must always take the difference of the two values found by substituting two values of the elementary variable quantities; and this rule being general, it supersedes the necessity of introducing a constant correction of the fluent in each particular case.

7. Particular values of fluents, limited on both sides, are distinguished by accents, $\backslash int\text{'}$.

Exception. In the case $n\; =\; -1$, the theorem fails, and we must substitute

x being the initial value of y, this theorem gives the increment of y corresponding to any increment of x beginning at the same time: it may be called the master key which opens a way to all the treasures of analysis. From Taylor, Meth. Incr.

Put $a+bx+cx^2=y$, $4ac-b^2=k$.

Fluents. 88. $\backslash int\; \backslash frac\{dx\}\{x^2(a+bx^2)^2\}\; =\; \backslash left(\; -\backslash frac\{1\}\{ax\}\; -\; \backslash frac\{4bx^2\}\{3a^2\}\; \backslash right)\; \backslash frac\{1\}\{a+bx^2\}\; -\; \backslash frac\{4b\}\{3a^2\}$
$\backslash int\; \backslash frac\{x\; dx\}\{a+bx^2\}$

89. $\backslash int\; \backslash frac\{dx\}\{x^3(a+bx^2)^2\}\; =\; \backslash left(\; -\backslash frac\{1\}\{2ax^2\}\; -\; \backslash frac\{5bx\}\{ba^2\}\; \backslash right)\; \backslash frac\{1\}\{a+bx^2\}\; -$
$\backslash frac\{5b\}\{3a^2\}\; \backslash int\; \backslash frac\{dx\}\{a+bx^2\}$

E. $x^\{N-1\}(a+bx^N)^\{-1\}\; dx$

a. $N=2r+1$, an odd number; putting $k^N\; =\; \backslash frac\{a\}\{b\}$,

$\backslash theta\; =\; \backslash frac\{180^\backslash circ\}\{N\},\; \backslash frac\{540^\backslash circ\}\{N\},\; \backslash frac\{900^\backslash circ\}\{N\},\; \backslash dots,\; \backslash frac\{(N-2)180^\backslash circ\}\{N\}$, $r$ values.

90. $\backslash int\; \backslash frac\{x^\{2r+1-m\}\; dx\}\{a+bx^\{2r+1\}\}\; =\; \backslash frac\{1\}\{nb(-k)^\{m-1\}\}\; \backslash text\{hl\}(x+k)\; +$
$\backslash frac\{1\}\{nbk^\{m-1\}\}$

$\backslash Sigma\; \backslash left(\; \backslash cos(m-1)\backslash theta\; \backslash text\{hl\}(x^2-2kx\; \backslash cos\; \backslash theta\; +\; k^2)\; +\; 2\; \backslash sin(m-1)\backslash theta\; \backslash text\{arctang\}\; \backslash frac\{x\; \backslash sin\; \backslash theta\}\{k-x\; \backslash cos\; \backslash theta\}\; \backslash right)$; the characteristic $\backslash Sigma$ implying the sum of the $r$ values depending on those of $\backslash theta$. From Cotes's discoveries.

91. $\backslash int\; \backslash frac\{x^\{2r-m\}\; dx\}\{x+bx^\{2r\}\}$; 1, when $\backslash frac\{a\}\{b\}$ is negative, putting $k^N\; =\; -\backslash frac\{a\}\{b\}$; $\backslash frac\{1\}\{nbk^\{m-1\}\}\; \backslash text\{hl\}(x-k)\; +\; \backslash frac\{1\}\{nb(-k)^\{m-1\}\}\; \backslash text\{hl\}(x+k)\; +\; \backslash frac\{1\}\{nbk^\{m-1\}\}\; \backslash Sigma\; \backslash left(\; \backslash cos(m-1)\backslash theta\; \backslash text\{hl\}(x^2-2kx\; \backslash cos\; \backslash theta\; +\; k^2)\; +\; 2\; \backslash sin(m-1)\backslash theta\; \backslash text\{arctang\}\; \backslash frac\{x\; \backslash sin\; \backslash theta\}\{k-x\; \backslash cos\; \backslash theta\}\; \backslash right)$; $\backslash Sigma$ relating to the $r-1$ values of

$\backslash theta,\; \backslash frac\{360^\backslash circ\}\{N\},\; \backslash frac\{720^\backslash circ\}\{N\},\; \backslash frac\{1080^\backslash circ\}\{N\},\; \backslash dots,\; \backslash frac\{(N-2)180^\backslash circ\}\{N\}$; 2, when $\backslash frac\{a\}\{b\}$ is positive, putting $k^N\; =\; \backslash frac\{a\}\{b\}$; $\backslash frac\{1\}\{nbk^\{m-1\}\}$

$\backslash Sigma\; \backslash left(\; \backslash cos(m-1)\backslash theta\; \backslash text\{hl\}(x^2-2kx\; \backslash cos\; \backslash theta\; +\; k^2)\; +\; 2\; \backslash sin(m-1)\backslash theta\; \backslash text\{arctang\}\; \backslash frac\{x\; \backslash sin\; \backslash theta\}\{k-x\; \backslash cos\; \backslash theta\}\; \backslash right)$; $\backslash Sigma$ relating to the $r$ values of $\backslash theta,\; \backslash frac\{180^\backslash circ\}\{N\},\; \backslash frac\{540^\backslash circ\}\{N\},\; \backslash frac\{900^\backslash circ\}\{N\},\; \backslash dots$

F. $x^\{2N-m\}(a+bx^N+cx^\{2N\})^\{-1\}\; dx$

92. $\backslash int\; \backslash frac\{x^\{2N-m\}\; dx\}\{a+bx^N+cx^\{2N\}\}$; first, $4ac$ being $>\; b^2$, and

putting $\backslash cos\; x\; =\; -\backslash frac\{b\}\{2\backslash sqrt\{ac\}\}$ and $\backslash theta\; =\; \backslash frac\{x\}\{N\}$,
$\backslash frac\{360^\backslash circ+x\}\{N\},\; \backslash dots,\; \backslash frac\{(N-1)360^\backslash circ+x\}\{N\}$; $\backslash frac\{\backslash cos\; x\}\{2Nck^\{m-1\}\}$

$\backslash Sigma\; \backslash left(\; -\backslash sin(m-N-1)\backslash theta\; \backslash text\{hl\}(x^2-2kx\; \backslash cos\; \backslash theta\; +\; k^2)\; +\; 2\; \backslash cos(m-N-1)\backslash theta\; \backslash text\{arctang\}\; \backslash frac\{x\; \backslash sin\; \backslash theta\}\{k-x\; \backslash cos\; \backslash theta\}\; \backslash right)$; secondly, $4ac$ being , and putting $\backslash sqrt\{b^2-4ac\}\; =\; h,\; \backslash frac\{b-h\}\{2\}\; =\; f,\; \backslash frac\{b+h\}\{2\}\; =\; g$,

$\backslash int\; \backslash frac\{x^m\; dx\}\{a+bx^N+cx^\{2N\}\}\; =\; \backslash frac\{c\}\{h\}\; \backslash left(\; \backslash int\; \backslash frac\{x^m\; dx\}\{cx^N+f\}\; -\; \backslash int\; \backslash frac\{x^m\; dx\}\{cx^N+g\}\; \backslash right)$

G. $x^m(x+f)^\{-1\}(x+g)^\{-1\}\; \backslash dots\; (x^2+ax+b)^\{-1\}\; \backslash dots\; dx$

93. $\backslash int\; \backslash frac\{dx\}\{(x+f)(x+g)\}\; =\; \backslash frac\{1\}\{g-f\}\; \backslash text\{hl\}\; \backslash frac\{x+f\}\{x+g\}$

94. $\backslash int\; \backslash frac\{x\; dx\}\{(x+f)(x+g)\}\; =\; \backslash frac\{1\}\{g-f\}\; (g\; \backslash text\{hl\}(x+g)\; -\; f\; \backslash text\{hl\}[x+f])$

95. $\backslash int\; \backslash frac\{dx\}\{(x+f)(x+g)(x+h)\}\; =\; \backslash frac\{1\}\{(g-f)(h-f)\}\; \backslash text\{hl\}(x+f)\; +\; \backslash frac\{1\}\{(f-g)(h-g)\}\; \backslash text\{hl\}(x+g)\; +\; \backslash frac\{1\}\{(f-h)(g-h)\}\; \backslash text\{hl\}(x+h)$

96. $\backslash int\; \backslash frac\{dx\}\{(x+f)(x^2+a)\}\; =\; \backslash frac\{1\}\{f^2+a\}\; \backslash left(\; \backslash text\{hl\}\; \backslash frac\{x+f\}\{\backslash sqrt\{x^2+a\}\}\; +\; f\; \backslash int\; \backslash frac\{dx\}\{x^2+a\}\; \backslash right)$

97. $\backslash int\; \backslash frac\{dx\}\{(x^2+a)(x^2+b)\}\; =\; \backslash frac\{1\}\{b-a\}\; \backslash left(\; \backslash int\; \backslash frac\{dx\}\{x^2+a\}\; -\; \backslash int\; \backslash frac\{dx\}\{x^2+b\}\; \backslash right)$

98. $\backslash int\; \backslash frac\{dx\}\{(x+f)(x^2+ax+b)\}\; =\; \backslash frac\{1\}\{f^2-af+b\}\; \backslash left(\; \backslash frac\{1\}\{2\}\; \backslash text\{hl\}\; \backslash frac\{x+f^2\}\{x^2+ax+b\}\; +\; f\; -\; \backslash frac\{1\}\{2\}a\; \backslash right)\; \backslash int\; \backslash frac\{dx\}\{x^2+ax+b\}$

H. $x^m(A+Bx+Cx^2\; \backslash dots)(a+bx+cx^2\; \backslash dots)^\{-1\}\; dx$

99. $\backslash int\; \backslash frac\{A+Bx+Cx^2\; \backslash dots\}\{a+bx+cx^2\; \backslash dots\}\; dx\; =\; \backslash Sigma\; \backslash text\{hl\}(x-\backslash xi)\; \backslash frac\{v\}\{\backslash xi\}$; $\backslash xi$ being successively each of the roots of the equation $a+bx+cx^2\; \backslash dots\; =\; 0,\; v\; =\; A+B\backslash xi+C\backslash xi^2\; \backslash dots$, and $\backslash xi\; =\; b+2c\backslash xi+3d\backslash xi^2\; \backslash dots$; provided, however, that the denominator contain higher powers of $x$ than the numerator, and that all the values of $\backslash xi$ be different; a limitation first laid down by Newton.

100. $\backslash int\; \backslash frac\{A+Bx+Cx^2\; \backslash dots\}\{a+bx+cx^2\; \backslash dots\}\; x^m\; dx\; =\; \backslash Sigma\; \backslash int\; \backslash frac\{x^m\; dx\}\{x-\backslash xi\}\; \backslash frac\{v\}\{\backslash xi\}$

101. $\backslash int\; \backslash frac\{A+Bx+Cx^2\; \backslash dots\}\{a+bx+cx^2\; \backslash dots\}\; \backslash frac\{dx\}\{x^m\}\; =\; \backslash Sigma\; \backslash int\; \backslash frac\{dx\}\{x^m(x-\backslash xi)\}\; \backslash frac\{v\}\{\backslash xi\}$

$\backslash operatorname\{tang\}\; \backslash frac\{\backslash sqrt\{y\}\}\{\backslash sqrt\{a\}\}$. The ambiguity of the roots being decided by the conditions of the problem.

Remark. In some of these cases, the signs of the roots, being ambiguous, require to be determined by the conditions of the problem; but we must adhere to the same root in the same solution.

Particular values, from $x=0$ to $x=a$; putting $\backslash pi\; =\; 3.14159$.

Particular values, from $x=0$ to $x=a$; putting $\backslash pi\; =\; 3.14159$.

Fluents. 212. $\backslash int\; x^3(a+bx^2)^\{\backslash frac\{5\}\{2\}\}\; dx\; =\; \backslash left(\; \backslash frac\{x^2\}\{9b\}\; -\; \backslash frac\{2a\}\{63b^2\}\; \backslash right)\; y^\{\backslash frac\{7\}\{2\}\}$

Particular values, from $x\; =\; 0$ to $x\; =\; a$; putting $\backslash pi\; =\; 3.14159$.

i. $\backslash int\; (a^2-x^2)^\{\backslash frac\{5\}\{2\}\}\; dx\; =\; \backslash frac\{5\backslash pi\; a^2\}\{32\}$

ii. $\backslash int\; x(a^2-x^2)^\{\backslash frac\{5\}\{2\}\}\; dx\; =\; \backslash frac\{a^7\}\{7\}$

iii. $\backslash int\; x^2(a^2-x^2)^\{\backslash frac\{5\}\{2\}\}\; dx\; =\; \backslash frac\{1\}\{8\}\; \backslash cdot\; \backslash frac\{5\backslash pi\; a^3\}\{32\}$

iv. $\backslash int\; x^3(a^2-x^2)^\{\backslash frac\{5\}\{2\}\}\; dx\; =\; \backslash frac\{2\}\{9\}\; \backslash cdot\; \backslash frac\{a^9\}\{7\}$

v. $\backslash int\; x^4(a^2-x^2)^\{\backslash frac\{5\}\{2\}\}\; dx\; =\; \backslash frac\{1\}\{8\}\; \backslash cdot\; \backslash frac\{3\}\{10\}\; \backslash cdot\; \backslash frac\{5\backslash pi\; a^\{10\}\}\{32\}$

vi. $\backslash int\; x^5(a^2-x^2)^\{\backslash frac\{5\}\{2\}\}\; dx\; =\; \backslash frac\{2\}\{9\}\; \backslash cdot\; \backslash frac\{4\}\{11\}\; \backslash cdot\; \backslash frac\{a^\{11\}\}\{7\}$

m. $x^\{-m\}(a+bx^2)^\{\backslash frac\{5\}\{2\}\}\; dx\; =\; x^\{-m\}y^\{\backslash frac\{5\}\{2\}\}dx$

213. $\backslash int\; \backslash frac\{(a+bx^2)^\{\backslash frac\{5\}\{2\}\}\}\{x\}\; dx\; =\; \backslash left(\; \backslash frac\{y^2\}\{5\}\; +\; \backslash frac\{ay\}\{3\}\; +\; a^2\; \backslash right)\; \backslash sqrt\{y\}\; +\; a^2\; \backslash int\; \backslash frac\{dx\}\{x\backslash sqrt\{y\}\}$

214. $\backslash int\; \backslash frac\{(a+bx^2)^\{\backslash frac\{5\}\{2\}\}\}\{x^2\}\; dx\; =\; -\backslash frac\{y^\{\backslash frac\{7\}\{2\}\}\}\{ax\}\; +\; \backslash frac\{6b\}\{a\}\; \backslash int\; y^\{\backslash frac\{5\}\{2\}\}\; dx$

215. $\backslash int\; \backslash frac\{(a+bx^2)^\{\backslash frac\{5\}\{2\}\}\}\{x^3\}\; dx\; =\; -\backslash frac\{y^\{\backslash frac\{7\}\{2\}\}\}\{2ax^2\}\; +\; \backslash frac\{5b\}\{2a\}\; \backslash int\; \backslash frac\{y^\{\backslash frac\{5\}\{2\}\}\}\{x\}\; dx$

E. $x^m(ax+bx^2)^\{\backslash frac\{n\}\{2\}\}\; dx$

a. $x^m(ax+bx^2)^\{-\backslash frac\{1\}\{2\}\}\; dx\; =\; x^m\; y^\{-\backslash frac\{1\}\{2\}\}\; dx$

216. $\backslash int\; \backslash frac\{dx\}\{\backslash sqrt\{(ax+bx^2)\}\}\; =\; \backslash frac\{1\}\{\backslash sqrt\{b\}\}\; \backslash text\{hl\}\; \backslash frac\{\backslash sqrt\{y\}\; +\; x\backslash sqrt\{b\}\}\{\backslash sqrt\{y\}\; -\; x\backslash sqrt\{b\}\}\; =\; \backslash frac\{2\}\{\backslash sqrt\{-b\}\}\; \backslash text\{arc\; tang\}\; \backslash frac\{x\backslash sqrt\{-b\}\}\{\backslash sqrt\{y\}\}$; thus

$\backslash int\; \backslash frac\{dx\}\{\backslash sqrt\{(x^2+x)\}\}\; =\; \backslash pm\; \backslash text\{hl\}\; (2x+1\; \backslash pm\; 2\backslash sqrt\{y\})$

$\backslash int\; \backslash frac\{dx\}\{\backslash sqrt\{(x^2-x)\}\}\; =\; \backslash pm\; \backslash text\{hl\}\; (1-2x\; \backslash mp\; 2\backslash sqrt\{y\})$

217. $\backslash int\; \backslash frac\{xdx\}\{\backslash sqrt\{(ax+bx^2)\}\}\; =\; \backslash frac\{\backslash sqrt\{y\}\}\{b\}\; -\; \backslash frac\{a\}\{2b\}\; \backslash int\; \backslash frac\{dx\}\{\backslash sqrt\{y\}\}$

218. $\backslash int\; \backslash frac\{x^2\; dx\}\{\backslash sqrt\{(ax+bx^2)\}\}\; =\; \backslash left(\; \backslash frac\{x\}\{2b\}\; -\; \backslash frac\{3a\}\{4b^2\}\; \backslash right)\; \backslash sqrt\{y\}\; +\; \backslash frac\{3a^2\}\{8b^2\}\; \backslash int\; \backslash frac\{dx\}\{\backslash sqrt\{y\}\}$

219. $\backslash int\; \backslash frac\{x^3\; dx\}\{\backslash sqrt\{(ax+bx^2)\}\}\; =\; \backslash left(\; \backslash frac\{x^2\}\{3b\}\; -\; \backslash frac\{5ax\}\{12b^2\}\; +\; \backslash frac\{5a^2\}\{8b^3\}\; \backslash right)\; \backslash sqrt\{y\}\; -\; \backslash frac\{5a^3\}\{16b^3\}\; \backslash int\; \backslash frac\{dx\}\{\backslash sqrt\{y\}\}$

b. $\backslash frac\{dx\}\{x^m\; \backslash sqrt\{(ax+bx^2)\}\}\; =\; \backslash frac\{dx\}\{x^m\; y^\{\backslash frac\{1\}\{2\}\}\}$

220. $\backslash int\; \backslash frac\{dx\}\{x\; \backslash sqrt\{(ax+bx^2)\}\}\; =\; -\backslash frac\{2\backslash sqrt\{y\}\}\{ax\}$

221. $\backslash int\; \backslash frac\{dx\}\{x^2\; \backslash sqrt\{(ax+bx^2)\}\}\; =\; \backslash left(\; -\backslash frac\{1\}\{3ax^2\}\; +\; \backslash frac\{2b\}\{3a^2x\}\; \backslash right)\; 2\backslash sqrt\{y\}$

222. $\backslash int\; \backslash frac\{dx\}\{x^2\; \backslash sqrt\{(ax+bx^2)\}\}\; =\; \backslash left(\; -\backslash frac\{1\}\{5ax^2\}\; +\; \backslash frac\{4b\}\{15a^2x^2\}\; -\; \backslash frac\{8b^2\}\{15a^3x\}\; \backslash right)\; 2\backslash sqrt\{y\}$

c. $x^m(ax+bx^2)^\{-\backslash frac\{5\}\{2\}\}\; dx$

223. $\backslash int\; \backslash frac\{dx\}\{(ax+bx^2)^\{\backslash frac\{5\}\{2\}\}\}\; =\; -\backslash frac\{2(2bx+a)\}\{a^2\; \backslash sqrt\{y\}\}$

224. $\backslash int\; \backslash frac\{xdx\}\{(ax+bx^2)^\{\backslash frac\{5\}\{2\}\}\}\; =\; \backslash frac\{2x\}\{a\backslash sqrt\{y\}\}$

225. $\backslash int\; \backslash frac\{x^2\; dx\}\{(ax+bx^2)^\{\backslash frac\{5\}\{2\}\}\}\; =\; -\backslash frac\{2x\}\{\backslash sqrt\{y\}\}\; +\; \backslash frac\{1\}\{b\}\; \backslash int\; \backslash frac\{dx\}\{\backslash sqrt\{y\}\}$

226. $\backslash int\; \backslash frac\{x^3\; dx\}\{(ax+bx^2)^\{\backslash frac\{5\}\{2\}\}\}\; =\; \backslash left(\; \backslash frac\{x^2\}\{b\}\; +\; \backslash frac\{3ax\}\{b^2\}\; \backslash right)\; \backslash frac\{1\}\{\backslash sqrt\{y\}\}\; -\; \backslash frac\{3a\}\{2b^2\}\; \backslash int\; \backslash frac\{dx\}\{\backslash sqrt\{y\}\}$

d. $\backslash frac\{dx\}\{x^m(ax+bx^2)^\{\backslash frac\{5\}\{2\}\}\}\; =\; \backslash frac\{dx\}\{x^m\; y^\{\backslash frac\{5\}\{2\}\}\}$

227. $\backslash int\; \backslash frac\{dx\}\{x(x+bx^2)^\{\backslash frac\{5\}\{2\}\}\}\; =\; -\backslash frac\{2\}\{3ax\backslash sqrt\{y\}\}\; -\; \backslash frac\{4b\}\{3a\}\; \backslash int\; \backslash frac\{dx\}\{y^\{\backslash frac\{5\}\{2\}\}\}$

228. $\backslash int\; \backslash frac\{dx\}\{x^2(ax+bx^2)^\{\backslash frac\{5\}\{2\}\}\}\; =\; \backslash left(\; -\backslash frac\{1\}\{5ax^2\}\; +\; \backslash frac\{2b\}\{5a^2x\}\; \backslash right)\; \backslash frac\{2\}\{\backslash sqrt\{y\}\}\; +\; \backslash frac\{8b^2\}\{5a^2\}\; \backslash int\; \backslash frac\{dx\}\{y^\{\backslash frac\{5\}\{2\}\}\}$

229. $\backslash int\; \backslash frac\{dx\}\{x^3(ax+bx^2)^\{\backslash frac\{5\}\{2\}\}\}\; =\; \backslash left(\; =\; \backslash frac\{1\}\{7ax^3\}\; +\; \backslash frac\{8b\}\{35a^2x^2\}\; -\; \backslash frac\{16b^2\}\{35a^3x\}\; \backslash right)\; \backslash frac\{2\}\{\backslash sqrt\{y\}\}\; -\; \backslash frac\{64b^3\}\{35a^3\}\; \backslash int\; \backslash frac\{dx\}\{y^\{\backslash frac\{5\}\{2\}\}\}$

e. $x^m(ax+bx^2)^\{-\backslash frac\{5\}\{2\}\}\; dx\; =\; x^m\; y^\{-\backslash frac\{5\}\{2\}\}\; dx$

230. $\backslash int\; \backslash frac\{dx\}\{(ax+bx^2)^\{\backslash frac\{5\}\{2\}\}\}\; =\; \backslash left(\; -\backslash frac\{2\}\{3y\}\; +\; \backslash frac\{16b\}\{3a^2\}\; \backslash right)\; \backslash frac\{2bx+a\}\{a^2\; \backslash sqrt\{y\}\}$

231. $\backslash int\; \backslash frac\{xdx\}\{(ax+bx^2)^\{\backslash frac\{5\}\{2\}\}\}\; =\; \backslash frac\{2x\}\{3ay\backslash sqrt\{y\}\}\; -\; \backslash frac\{8(2bx+a)\}\{3a^2\; \backslash sqrt\{y\}\}\; =\; \backslash left(\; \backslash frac\{1\}\{a+bx\}\; -\; \backslash frac\{4(2bx+a)\}\{a^2\}\; \backslash right)\; \backslash frac\{2\}\{a\backslash sqrt\{y\}\}$

232. $\backslash int\; \backslash frac\{x^2\; dx\}\{(ax+bx^2)^\{\backslash frac\{5\}\{2\}\}\}\; =\; \backslash left(\; \backslash frac\{2x^2\}\{3ay\}\; +\; \backslash frac\{4x\}\{3a^2\}\; \backslash right)\; \backslash frac\{1\}\{\backslash sqrt\{y\}\}$

233. $\backslash int\; \backslash frac\{x^3\; dx\}\{(ax+bx^2)^\{\backslash frac\{5\}\{2\}\}\}\; =\; \backslash frac\{2x^3\}\{3ay\backslash sqrt\{y\}\}$

f. $\backslash frac\{dx\}\{x^m(ax+bx^2)^\{\backslash frac\{5\}\{2\}\}\}\; =\; \backslash frac\{dx\}\{x^m\; y^\{\backslash frac\{5\}\{2\}\}\}$

234. $\backslash int\; \backslash frac\{dx\}\{x(ax+bx^2)^\{\backslash frac\{5\}\{2\}\}\}\; =\; -\backslash frac\{2\}\{5axy\backslash sqrt\{-y\}\}\; -\; \backslash frac\{8b\}\{5a\}\; \backslash int\; \backslash frac\{dx\}\{y^\{\backslash frac\{5\}\{2\}\}\}$

235. $\backslash int\; \backslash frac\{dx\}\{x^2(ax+bx^2)^\{\backslash frac\{5\}\{2\}\}\}\; =\; \backslash left(\; -\backslash frac\{1\}\{7ax^2\}\; +\; \backslash frac\{2b\}\{7a^2x\}\; \backslash right)\; \backslash frac\{2\}\{y\backslash sqrt\{y\}\}\; +\; \backslash frac\{16b^2\}\{7a^2\}\; \backslash int\; \backslash frac\{dx\}\{y^\{\backslash frac\{5\}\{2\}\}\}$

236. $\backslash int\; \backslash frac\{dx\}\{x^3(ax+bx^2)^\{\backslash frac\{5\}\{2\}\}\}\; =\; \backslash left(\; -\backslash frac\{1\}\{9ax^3\}\; +\; \backslash frac\{4b\}\{21a^2x^2\}\; -\; \backslash frac\{8b^2\}\{21a^3x\}\; \backslash right)\; \backslash frac\{2\}\{y\backslash sqrt\{y\}\}\; -\; \backslash frac\{64b^3\}\{21a^3\}\; \backslash int\; \backslash frac\{dx\}\{y^\{\backslash frac\{5\}\{2\}\}\}$

(A): thus

Particular values, from $x\; =\; 0$ to $x\; =\; a$.

Relation of particular values, from $x=0$ to $x=1$

Particular value, from $x=0$ to $x=1$

SECTION V.—Circular Fluxions.

It may be remarked that $\backslash cos^N\; \backslash phi\; =\; \backslash frac\{1\}\{2^\{N-1\}\}$

$(\backslash cos\; N\backslash phi\; +\; N\; \backslash cos\; (N-2)\backslash phi\; +\; N\; \backslash frac\{N-1\}\{2\}\; \backslash cos\; (N-4)\backslash phi\; +\; \backslash dots)$; continuing the series through all positive angles, and putting $\backslash frac\{1\}\{2\}$ instead of $\backslash cos\; 0$.

We have for the powers of $\backslash sin\; \backslash phi$, $\backslash sin^N\; \backslash phi\; =\; \backslash frac\{1\}\{2^\{N-1\}\}$ $(\backslash cos\; N\backslash phi\; -\; N\; \backslash cos\; (N-2)\backslash phi\; +\; N\; \backslash frac\{N-1\}\{2\}\; \backslash cos\; (N-4)\backslash phi\; -\; \backslash dots)$; + when $N=4P$, - when

$N=4P+2$; and $\backslash sin^N\; \backslash phi\; =\; \backslash frac\{1\}\{2^\{N-1\}\}$ $(\backslash sin\; N\backslash phi$

$-\backslash sin(N-2)\backslash phi\; +\; N\; \backslash frac\{N-1\}\{2\}\; \backslash sin(N-4)\backslash phi\; -\; \backslash dots$,
+ when $N=4r+1$, and - when $N=4r+3$; the
last term, when it becomes $\backslash cos\; 0$, being altered
to $\backslash frac\{1\}\{2\}$.

c. $\backslash sin^2\backslash phi\; \backslash cos^3\backslash phi\; d\backslash phi$

c. $\backslash sin^4\backslash phi\; \backslash cos^3\backslash phi\; d\backslash phi$

405. $\backslash int\; \backslash frac\{d\backslash phi\}\{\backslash cos^5\; \backslash phi\}\; =\; \backslash left(\; \backslash frac\{1\}\{4\; \backslash cos^4\; \backslash phi\}\; +\; \backslash frac\{3\}\{8\; \backslash cos^2\; \backslash phi\}\; \backslash right)\; \backslash sin\; \backslash phi\; +\; \backslash frac\{3\}\{8\}$
hl tang $\backslash left(\; 45^\backslash circ\; +\; \backslash frac\{\backslash phi\}\{2\}\; \backslash right)$

406. $\backslash int\; \backslash frac\{d\backslash phi\}\{\backslash cos^6\; \backslash phi\}\; =\; \backslash left(\; \backslash frac\{1\}\{5\; \backslash cos^5\; \backslash phi\}\; +\; \backslash frac\{4\}\{15\; \backslash cos^3\; \backslash phi\}\; +\; \backslash frac\{8\}\{15\; \backslash cos\; \backslash phi\}\; \backslash right)\; \backslash sin\; \backslash phi$
f. $\backslash sin^m\; \backslash phi\; \backslash cos^\{-n\}\; \backslash phi\; d\backslash phi$
a. $\backslash sin^m\; \backslash phi\; \backslash cos^\{-1\}\; \backslash phi\; d\backslash phi$

407. $\backslash int\; \backslash frac\{\backslash sin\; \backslash phi\; d\backslash phi\}\{\backslash cos\; \backslash phi\}\; =\; -\backslash text\{hl\}\; \backslash cos\; \backslash phi\; =\; \backslash text\{hl\}\; \backslash sec\; \backslash phi$

408. $\backslash int\; \backslash frac\{\backslash sin^2\; \backslash phi\; d\backslash phi\}\{\backslash cos\; \backslash phi\}\; =\; -\backslash sin\; \backslash phi\; +\; \backslash text\{hl\}\; \backslash text\{\; tang\}\; \backslash left(\; 45^\backslash circ\; +\; \backslash frac\{\backslash phi\}\{2\}\; \backslash right)$

409. $\backslash int\; \backslash frac\{\backslash sin^3\; \backslash phi\; d\backslash phi\}\{\backslash cos\; \backslash phi\}\; =\; -\backslash frac\{\backslash sin^2\; \backslash phi\}\{2\}\; -\; \backslash text\{hl\}\; \backslash cos\; \backslash phi$

410. $\backslash int\; \backslash frac\{\backslash sin^4\; \backslash phi\; d\backslash phi\}\{\backslash cos\; \backslash phi\}\; =\; -\backslash frac\{\backslash sin^3\; \backslash phi\}\{3\}\; -\; \backslash sin\; \backslash phi\; +\; \backslash text\{hl\}\; \backslash text\{\; tang\}\; \backslash left(\; 45^\backslash circ\; +\; \backslash frac\{\backslash phi\}\{2\}\; \backslash right)$

411. $\backslash int\; \backslash frac\{\backslash sin^5\; \backslash phi\; d\backslash phi\}\{\backslash cos\; \backslash phi\}\; =\; -\backslash frac\{\backslash sin^4\; \backslash phi\}\{4\}\; -\; \backslash frac\{\backslash sin^2\; \backslash phi\}\{2\}\; -\; \backslash text\{hl\}\; \backslash cos\; \backslash phi$

412. $\backslash int\; \backslash frac\{\backslash sin^6\; \backslash phi\; d\backslash phi\}\{\backslash cos\; \backslash phi\}\; =\; -\backslash frac\{\backslash sin^5\; \backslash phi\}\{5\}\; -\; \backslash frac\{\backslash sin^3\; \backslash phi\}\{3\}\; -\; \backslash sin\; \backslash phi\; +\; \backslash text\{hl\}\; \backslash text\{\; tang\}\; \backslash left(\; 45^\backslash circ\; +\; \backslash frac\{\backslash phi\}\{2\}\; \backslash right)$
b. $\backslash sin^m\; \backslash phi\; \backslash cos^\{-2\}\; \backslash phi\; d\backslash phi$

413. $\backslash int\; \backslash frac\{\backslash sin\; \backslash phi\; d\backslash phi\}\{\backslash cos^2\; \backslash phi\}\; =\; \backslash frac\{1\}\{\backslash cos\; \backslash phi\}\; =\; \backslash sec\; \backslash phi$

414. $\backslash int\; \backslash frac\{\backslash sin^2\; \backslash phi\; d\backslash phi\}\{\backslash cos^2\; \backslash phi\}\; =\; \backslash frac\{\backslash sin\; \backslash phi\}\{\backslash cos\; \backslash phi\}\; =\; \backslash tan\; \backslash phi$

415. $\backslash int\; \backslash frac\{\backslash sin^3\; \backslash phi\; d\backslash phi\}\{\backslash cos^2\; \backslash phi\}\; =\; \backslash left(\; -\backslash sin^2\; \backslash phi\; +\; 2\; \backslash right)\; \backslash frac\{1\}\{\backslash cos\; \backslash phi\}\; =\; \backslash cos\; \backslash phi\; +\; \backslash sec\; \backslash phi$

416. $\backslash int\; \backslash frac\{\backslash sin^4\; \backslash phi\; d\backslash phi\}\{\backslash cos^2\; \backslash phi\}\; =\; \backslash left(\; -\backslash frac\{1\}\{2\}\; \backslash sin^3\; \backslash phi\; +\; \backslash frac\{3\}\{2\}\; \backslash sin\; \backslash phi\; \backslash right)\; \backslash frac\{1\}\{\backslash cos\; \backslash phi\}\; =\; \backslash frac\{3\}\{2\}\; \backslash phi$

417. $\backslash int\; \backslash frac\{\backslash sin^5\; \backslash phi\; d\backslash phi\}\{\backslash cos^2\; \backslash phi\}\; =\; \backslash left(\; -\backslash frac\{1\}\{3\}\; \backslash sin^4\; \backslash phi\; -\; \backslash frac\{4\}\{3\}\; \backslash sin^2\; \backslash phi\; +\; \backslash frac\{8\}\{3\}\; \backslash right)\; \backslash frac\{1\}\{\backslash cos\; \backslash phi\}$

418. $\backslash int\; \backslash frac\{\backslash sin^6\; \backslash phi\; d\backslash phi\}\{\backslash cos^2\; \backslash phi\}\; =\; \backslash left(\; -\backslash frac\{1\}\{4\}\; \backslash sin^5\; \backslash phi\; -\; \backslash frac\{5\}\{8\}\; \backslash sin^3\; \backslash phi\; +\; \backslash frac\{15\}\{8\}\; \backslash sin\; \backslash phi\; \backslash right)\; \backslash frac\{1\}\{\backslash cos\; \backslash phi\}\; =\; \backslash frac\{15\}\{8\}\; \backslash phi$

c. $\backslash sin^m\; \backslash phi\; \backslash cos^\{-3\}\; \backslash phi\; d\backslash phi$

419. $\backslash int\; \backslash frac\{\backslash sin\; \backslash phi\; d\backslash phi\}\{\backslash cos^3\; \backslash phi\}\; =\; \backslash frac\{1\}\{2\; \backslash cos^2\; \backslash phi\}$

420. $\backslash int\; \backslash frac\{\backslash sin^2\; \backslash phi\; d\backslash phi\}\{\backslash cos^3\; \backslash phi\}\; =\; \backslash frac\{\backslash sin\; \backslash phi\}\{2\; \backslash cos^2\; \backslash phi\}\; =\; \backslash frac\{1\}\{2\}\; \backslash text\{hl\}\; \backslash text\{\; tang\}\; \backslash left(\; 45^\backslash circ\; +\; \backslash frac\{\backslash phi\}\{2\}\; \backslash right)$

421. $\backslash int\; \backslash frac\{\backslash sin^3\; \backslash phi\; d\backslash phi\}\{\backslash cos^3\; \backslash phi\}\; =\; \backslash frac\{1\}\{2\; \backslash cos^2\; \backslash phi\}\; +\; \backslash text\{hl\}\; \backslash cos\; \backslash phi$

422. $\backslash int\; \backslash frac\{\backslash sin^4\; \backslash phi\; d\backslash phi\}\{\backslash cos^3\; \backslash phi\}\; =\; \backslash left(\; -\backslash sin^3\; \backslash phi\; +\; \backslash frac\{3\}\{2\}\; \backslash sin\; \backslash phi\; \backslash right)\; \backslash frac\{1\}\{\backslash cos^2\; \backslash phi\}\; =\; \backslash frac\{3\}\{2\}\; \backslash text\{hl\}$
tang $\backslash left(\; 45^\backslash circ\; +\; \backslash frac\{\backslash phi\}\{2\}\; \backslash right)$

423. $\backslash int\; \backslash frac\{\backslash sin^5\; \backslash phi\; d\backslash phi\}\{\backslash cos^3\; \backslash phi\}\; =\; \backslash left(\; -\backslash frac\{1\}\{2\}\; \backslash sin^4\; \backslash phi\; +\; 1\; \backslash right)\; \backslash frac\{1\}\{\backslash cos^2\; \backslash phi\}\; +\; 2\; \backslash text\{hl\}\; \backslash cos\; \backslash phi$

424. $\backslash int\; \backslash frac\{\backslash sin^6\; \backslash phi\; d\backslash phi\}\{\backslash cos^3\; \backslash phi\}\; =\; \backslash left(\; -\backslash frac\{1\}\{3\}\; \backslash sin^5\; \backslash phi\; -\; \backslash frac\{5\}\{3\}\; \backslash sin^3\; \backslash phi\; +\; \backslash frac\{5\}\{2\}\; \backslash sin\; \backslash phi\; \backslash right)\; \backslash frac\{1\}\{\backslash cos^2\; \backslash phi\}\; =\; \backslash frac\{5\}\{2\}\; \backslash text\{hl\}\; \backslash text\{\; tang\}\; \backslash left(\; 45^\backslash circ\; +\; \backslash frac\{\backslash phi\}\{2\}\; \backslash right)$
d. $\backslash sin^m\; \backslash phi\; \backslash cos^\{-4\}\; \backslash phi\; d\backslash phi$

425. $\backslash int\; \backslash frac\{\backslash sin\; \backslash phi\; d\backslash phi\}\{\backslash cos^4\; \backslash phi\}\; =\; \backslash frac\{1\}\{3\; \backslash cos^3\; \backslash phi\}$

426. $\backslash int\; \backslash frac\{\backslash sin^2\; \backslash phi\; d\backslash phi\}\{\backslash cos^4\; \backslash phi\}\; =\; \backslash frac\{\backslash sin^3\; \backslash phi\}\{3\; \backslash cos^3\; \backslash phi\}\; =\; \backslash frac\{1\}\{3\}\; \backslash text\{tang\}^3\; \backslash phi$

427. $\backslash int\; \backslash frac\{\backslash sin^3\; \backslash phi\; d\backslash phi\}\{\backslash cos^4\; \backslash phi\}\; =\; \backslash left(\; \backslash sin^2\; \backslash phi\; -\; \backslash frac\{2\}\{3\}\; \backslash right)\; \backslash frac\{1\}\{\backslash cos^3\; \backslash phi\}$

428. $\backslash int\; \backslash frac\{\backslash sin^4\; \backslash phi\; d\backslash phi\}\{\backslash cos^4\; \backslash phi\}\; =\; \backslash left(\; \backslash frac\{4\}\{3\}\; \backslash sin^3\; \backslash phi\; -\; \backslash sin\; \backslash phi\; \backslash right)\; \backslash frac\{1\}\{\backslash cos^3\; \backslash phi\}\; =\; \backslash frac\{1\}\{3\}\; \backslash text\{tang\}^3\; \backslash phi\; -\; \backslash text\{tang\}\; \backslash phi\; +\; \backslash phi$

429. $\backslash int\; \backslash frac\{\backslash sin^5\; \backslash phi\; d\backslash phi\}\{\backslash cos^4\; \backslash phi\}\; =\; \backslash left(\; -\backslash sin^4\; \backslash phi\; +\; \backslash frac\{4\}\{3\}\; \backslash sin^2\; \backslash phi\; -\; \backslash frac\{8\}\{3\}\; \backslash right)\; \backslash frac\{1\}\{\backslash cos^3\; \backslash phi\}$

430. $\backslash int\; \backslash frac\{\backslash sin^6\; \backslash phi\; d\backslash phi\}\{\backslash cos^4\; \backslash phi\}\; =\; \backslash left(\; -\backslash frac\{1\}\{2\}\; \backslash sin^5\; \backslash phi\; +\; \backslash frac\{10\}\{3\}\; \backslash sin^3\; \backslash phi\; -\; \backslash frac\{5\}\{2\}\; \backslash sin\; \backslash phi\; \backslash right)\; \backslash frac\{1\}\{\backslash cos^3\; \backslash phi\}\; =\; \backslash frac\{5\}\{2\}\; \backslash phi$

e. $\backslash sin^m\; \backslash phi\; \backslash cos^\{-5\}\; \backslash phi\; d\backslash phi$

431. $\backslash int\; \backslash frac\{\backslash sin\; \backslash phi\; d\backslash phi\}\{\backslash cos^5\; \backslash phi\}\; =\; \backslash frac\{1\}\{4\; \backslash cos^4\; \backslash phi\}$

432. $\backslash int\; \backslash frac\{\backslash sin^2\; \backslash phi\; d\backslash phi\}\{\backslash cos^5\; \backslash phi\}\; =\; \backslash left(\; \backslash frac\{1\}\{8\}\; \backslash sin^3\; \backslash phi\; +\; \backslash frac\{1\}\{8\}\; \backslash sin\; \backslash phi\; \backslash right)\; \backslash frac\{1\}\{\backslash cos^4\; \backslash phi\}\; =\; \backslash frac\{1\}\{8\}\; \backslash text\{hl\}$
tang $\backslash left(\; 45^\backslash circ\; +\; \backslash frac\{\backslash phi\}\{2\}\; \backslash right)$

433. $\backslash int\; \backslash frac\{\backslash sin^3\; \backslash phi\; d\backslash phi\}\{\backslash cos^5\; \backslash phi\}\; =\; \backslash frac\{\backslash sin^4\; \backslash phi\}\{4\; \backslash cos^4\; \backslash phi\}\; =\; \backslash frac\{1\}\{4\}\; \backslash text\{tang\}^4\; \backslash phi$

434. $\backslash int\; \backslash frac\{\backslash sin^4\; \backslash phi\; d\backslash phi\}\{\backslash cos^5\; \backslash phi\}\; =\; \backslash left(\; \backslash frac\{5\}\{8\}\; \backslash sin^3\; \backslash phi\; -\; \backslash frac\{3\}\{8\}\; \backslash sin\; \backslash phi\; \backslash right)\; \backslash frac\{1\}\{\backslash cos^4\; \backslash phi\}\; =\; \backslash frac\{3\}\{8\}\; \backslash text\{hl\}$
tang $\backslash left(\; 45^\backslash circ\; +\; \backslash frac\{\backslash phi\}\{2\}\; \backslash right)$

435. $\backslash int\; \backslash frac\{\backslash sin^5\; \backslash phi\; d\backslash phi\}\{\backslash cos^5\; \backslash phi\}\; =\; \backslash left(\; \backslash frac\{3\}\{4\}\; \backslash sin^4\; \backslash phi\; -\; \backslash frac\{1\}\{2\}\; \backslash sin^2\; \backslash phi\; \backslash right)\; \backslash frac\{1\}\{\backslash cos^4\; \backslash phi\}\; =\; \backslash text\{hl\}$
$\backslash cos\; \backslash phi\; =\; \backslash frac\{1\}\{4\}\; \backslash text\{tang\}^4\; \backslash phi\; -\; \backslash frac\{1\}\{2\}\; \backslash text\{tang\}^2\; \backslash phi\; -\; \backslash text\{hl\}\; \backslash cos\; \backslash phi$

436. $\backslash int\; \backslash frac\{\backslash sin^6\; \backslash phi\; d\backslash phi\}\{\backslash cos^5\; \backslash phi\}\; =\; \backslash left(\; -\backslash sin^5\; \backslash phi\; +\; \backslash frac\{25\}\{8\}\; \backslash sin^3\; \backslash phi\; -\; \backslash frac\{15\}\{8\}\; \backslash sin\; \backslash phi\; \backslash right)\; \backslash frac\{1\}\{\backslash cos^4\; \backslash phi\}\; =\; \backslash frac\{15\}\{8\}\; \backslash text\{hl\}\; \backslash text\{\; tang\}\; \backslash left(\; 45^\backslash circ\; +\; \backslash frac\{\backslash phi\}\{2\}\; \backslash right)$
f. $\backslash sin^m\; \backslash phi\; \backslash cos^\{-6\}\; \backslash phi\; d\backslash phi$

P. $(a+b\; \backslash cos\; \backslash phi)^\{-m\}\; (f+g\; \backslash cos\; \backslash phi)\; d\backslash phi$

or, for $a=b$

Q. $(a+b\; \backslash cos\; \backslash phi)^\{-1\}\; \backslash sin\; \backslash phi\; d\backslash phi$

R. $(1+a\; \backslash cos\; \backslash phi)^m\; d\backslash phi$. For fractional powers see Méc. Cél.; also Ivory and Wallace, Ed. Trans. 1798, 1805.

Since $\backslash int\; Y\; dZ\; =\; \backslash frac\{dY\}\{dX\}\; \backslash int\; Z\; dX\; -\; \backslash frac\{d^2\; Y\}\{dX^2\}\; \backslash int\; Z\; dX^2\; +\; \backslash dots$ (n5), taking the $dX$ of this theorem = $\backslash frac\{dx\}\{x\}$, $Y\; =\; hl^m\; x$ and $dZ\; =\; y\; dx$, we have

When $n$ is a negative whole number = $-n$, we may obtain a finite series by making $Y\; =\; yx$, and $dZ\; =\; \backslash frac\{dx\}\{x\}\; hl^\{-n\}\; x$, but the last term still contains a fluent.

In the theorem $\backslash int\; Y\; dZ\; =\; \backslash frac\{dY\}\{dX\}\; \backslash int\; Z\; dY\; -\; \backslash dots$

we may put either $dX\; =\; dx$, $Y\; =\; y$, and $dZ\; =\; a^x\; dx\; =\; d\; \backslash frac\{a^x\}\{hl\; a\}$, or $dX\; =\; dx$, $Y\; =\; a^x$, and $dZ\; =\; y\; dx$; and, in the former manner, we obtain,

Fluents. 560. $\backslash int\; a^x\; x^2\; dx\; =\; \backslash frac\{a^x\; x^3\}\{\backslash text\{hl\; \}\; a\}\; -\; \backslash frac\{3a^x\; x^2\}\{\backslash text\{hl\}^2\; a\}\; +\; \backslash frac\{3.2a^x\; x\}\{\backslash text\{hl\}^3\; a\}\; -\; \backslash frac\{3.2.1a^x\}\{\backslash text\{hl\}^4\; a\}$

561. $\backslash int\; \backslash frac\{a^x\; dx\}\{x\}\; =\; \backslash text\{hl\; \}\; x\; +\; \backslash frac\{x\; \backslash text\{hl\; \}\; a\}\{1\}\; +\; \backslash frac\{x^2\; \backslash text\{hl\}^2\; a\}\{1.2.2\}\; +\; \backslash frac\{x^3\; \backslash text\{hl\}^3\; a\}\{1.2.3.3\}\; +\; \backslash dots$

562. $\backslash int\; \backslash frac\{a^x\; dx\}\{x^2\}\; =\; -\backslash frac\{a^x\}\{x\}\; +\; \backslash text\{hl\; \}\; a\; \backslash int\; \backslash frac\{a^x\; dx\}\{x\}$

563. $\backslash int\; \backslash frac\{a^x\; dx\}\{x^3\}\; =\; -\backslash frac\{a^x\}\{2x^2\}\; -\; \backslash frac\{a^x\; \backslash text\{hl\; \}\; a\}\{2.1x\}\; +\; \backslash frac\{\backslash text\{hl\}^2\; a\}\{2.1\}\; \backslash int\; \backslash frac\{a^x\; dx\}\{x\}$

564. $\backslash int\; \backslash frac\{a^x\; dx\}\{x^4\}\; =\; -\backslash frac\{a^x\}\{3x^3\}\; -\; \backslash frac\{a^x\; \backslash text\{hl\; \}\; a\}\{3.2x^2\}\; -\; \backslash frac\{a^x\; \backslash text\{hl\}^2\; a\}\{3.2.1\}\; +\; \backslash frac\{\backslash text\{hl\}^3\; a\}\{3.2.1\}\; \backslash int\; \backslash frac\{a^x\; dx\}\{x\}$

565. $\backslash int\; \backslash frac\{a^x\; dx\}\{\backslash sqrt\{x\}\}\; =\; \backslash frac\{a^x\}\{\backslash sqrt\{x\}\}\; \backslash left(\; \backslash frac\{1\}\{\backslash text\{hl\; \}\; a\}\; -\; \backslash frac\{1\}\{2x\; \backslash text\{hl\}^2\; a\}\; +\; \backslash frac\{1.3\}\{(2x)^2\; \backslash text\{hl\}^3\; a\}\; +\; \backslash frac\{1.3.5\}\{(2x)^3\; \backslash text\{hl\}^4\; a\}\; +\; \backslash dots\; \backslash right)$

$=\; \backslash frac\{a^x\}\{\backslash sqrt\{x\}\}\; \backslash left(\; \backslash frac\{2x\}\{1\}\; -\; \backslash frac\{(2x)^2\; \backslash text\{hl\; \}\; a\}\{1.3\}\; +\; \backslash frac\{(2x)^3\; \backslash text\{hl\}^2\; a\}\{1.3.5\}\; -\; \backslash dots\; \backslash right)$

566. $\backslash int\; \backslash frac\{a^x\; dx\}\{1-x\}\; =\; a^x\; \backslash left(\; \backslash frac\{1\}\{(1-x)\; \backslash text\{hl\; \}\; a\}\; -\; \backslash frac\{1\}\{(1-x)^2\; \backslash text\{hl\}^2\; a\}\; +\; \backslash frac\{1.2\}\{(1-x)^3\; \backslash text\{hl\}^3\; a\}\; -\; \backslash frac\{1.2.5\}\{(1-x)^4\; \backslash text\{hl\}^4\; a\}\; +\; \backslash dots\; \backslash right)$

$B.\; a^\{mx\}\; x^n\; dx$

567. $\backslash int\; a^\{mx\}\; x^n\; dx\; =\; \backslash frac\{1\}\{m+n+1\}\; \backslash int\; a^y\; y^n\; dy$; making $y=mx$

$C.\; x^\{m+n\}\; dx$

568. $\backslash int\; x^\{m+n\}\; dx\; =\; \backslash int\; \backslash left(\; 1\; +\; \backslash frac\{nx\; \backslash text\{hl\; \}\; x\}\{1\}\; +\; \backslash frac\{n^2\; x^2\; \backslash text\{hl\}^2\; x\}\{1.2\}\; +\; \backslash frac\{n^3\; x^3\; \backslash text\{hl\}^3\; x\}\{1.2.3\}\; +\; \backslash dots\; \backslash right)\; x^m\; dx$

Particular value, from $x=0$ to $x=1$

569. $\backslash int\; x^\{m+n\}\; dx\; =\; \backslash frac\{1\}\{m+1\}\; -\; \backslash frac\{n\}\{(m+2)^2\}\; +\; \backslash frac\{n^2\}\{(m+3)^3\}\; -\; \backslash dots$

$D.\; e^\{-rx\}\; dx$

Putting, in the Taylorian theorem,

$\backslash int\; Y\; dZ\; =\; \backslash frac\{dY\}\{dX\}\; \backslash int\; Z\; dX\; -\; \backslash dots$ (n.5) $Y\; =\; e^\{-rx\}$, $Z\; =\; x$, and $dX\; =\; d(-rx)\; =\; -r\; dx$, we have,

570. $\backslash int\; e^\{-rx\}\; dx\; =\; e^\{-rx\}\; \backslash left(\; -\backslash frac\{2\}\{3\}\; x^3\; \backslash right)\; -\; e^\{-rx\}\; \backslash frac\{4\}\{3.5\}\; x^5\; +\; e^\{-rx\}\; \backslash left(\; -\backslash frac\{8\}\{3.5.7\}\; x^7\; \backslash right)\; -\; \backslash dots$

$=\; -e^\{-rx\}\; \backslash left(\; \backslash frac\{2\}\{3\}\; x^3\; +\; \backslash frac\{4\}\{3.5\}\; x^5\; +\; \backslash frac\{8\}\{3.5.7\}\; x^7\; +\; \backslash dots\; \backslash right)$

Particular value, from $x=-\backslash infty$ to $x=+\backslash infty$

571. $\backslash int\; e^\{-rx\}\; dx\; =\; \backslash sqrt\{3.14159\}$, Laplace, Méc. Cél. X.; or thus,

$\backslash int\; e^\{-rx\}\; dx\; =\; \backslash int\; y\; dx$, $-rx\; =\; \backslash text\{hl\; \}\; y$, $rx\; =\; \backslash text\{hl\; \}\; \backslash frac\{1\}\{y\}$, $x\; =\; \backslash frac\{\backslash text\{hl\; \}\; \backslash frac\{1\}\{y\}\}\{r\}$; $dy\; =\; -e^\{-rx\}\; 2x\; dx$; $\backslash frac\{dy\}\{\backslash text\{hl\; \}\; \backslash frac\{1\}\{y\}\}\; =\; -2e^\{-rx\}\; dx$, and

$\backslash int\; e^\{-rx\}\; dx\; =\; -\backslash frac\{1\}\{2\}\; \backslash int\; \backslash frac\{dy\}\{\backslash text\{hl\; \}\; \backslash frac\{1\}\{y\}\}$ (n.554): in this expression we may make $\backslash frac\{1\}\{\backslash text\{hl\; \}\; \backslash frac\{1\}\{y\}\}\; =\; Y$, then $dY\; =\; \backslash frac\{d\; \backslash text\{hl\; \}\; \backslash frac\{1\}\{y\}\}\{\backslash text\{hl\; \}\; \backslash frac\{1\}\{y\}\}$, and if we wish to have $\backslash frac\{dY\}\{dX\}\; =\; Y$,

we must take $dX\; =\; d\; \backslash text\{hl\; \}\; \backslash frac\{1\}\{y\}$, and $Z$ will be $=y$, so that the series will give us $\backslash int\; \backslash frac\{dy\}\{\backslash text\{hl\; \}\; \backslash frac\{1\}\{y\}\}\; =$

$\backslash frac\{1\}\{2(\backslash text\{hl\; \}\; \backslash frac\{1\}\{y\})^\{\backslash frac\{1\}\{2\}\}\}\; -\; \backslash frac\{3\}\{4(\backslash text\{hl\; \}\; \backslash frac\{1\}\{y\})^\{\backslash frac\{3\}\{2\}\}\}\; +\; \backslash frac\{5\}\{8(\backslash text\{hl\; \}\; \backslash frac\{1\}\{y\})^\{\backslash frac\{5\}\{2\}\}\}\; -\; \backslash dots\; =$

$\backslash frac\{y\}\{\backslash sqrt\{(\backslash text\{hl\; \}\; \backslash frac\{1\}\{y\})\}\}\; \backslash left(\; \backslash frac\{1\}\{2(\backslash text\{hl\; \}\; \backslash frac\{1\}\{y\})^\{\backslash frac\{1\}\{2\}\}\}\; -\; \backslash frac\{3\}\{4(\backslash text\{hl\; \}\; \backslash frac\{1\}\{y\})^\{\backslash frac\{3\}\{2\}\}\}\; +\; \backslash frac\{5\}\{8(\backslash text\{hl\; \}\; \backslash frac\{1\}\{y\})^\{\backslash frac\{5\}\{2\}\}\}\; -\; \backslash dots\; \backslash right)$; but these series will fail in the extreme cases, although they converge with sufficient rapidity in most others.

$E.\; \backslash int\; e^\{-mx\}\; dx$

Putting $e^\{x^m\}\; =\; Y$, we have $dY\; =\; e^\{x^m\}\; m\; x^\{m-1\}\; dx$ and taking $dX\; =\; x^\{m-1\}\; dx$, and $Z\; =\; x$, we obtain the series.

572. $\backslash int\; e^\{x^m\}\; dx\; =\; m\; e^\{x^m\}\; \backslash frac\{1\}\{m+1\}\; x^\{m+1\}\; m^m\; e^\{x^m\}$

$=\; \backslash frac\{1\}\{(m+1)(2m+1)\}\; x^\{2m+1\}\; +\; \backslash dots$

$F.\; e^\{ax\}\; \backslash sin^m\; x\; dx$

573. $\backslash int\; e^\{ax\}\; \backslash sin\; x\; dx\; =\; \backslash frac\{e^\{ax\}\; (a\; \backslash sin\; x\; -\; \backslash cos\; x)\}\{a^2\; +\; 1\}$

574. $\backslash int\; e^\{ax\}\; \backslash sin^2\; x\; dx\; =\; \backslash frac\{e^\{ax\}\; \backslash sin\; x\; (a\; \backslash sin\; x\; -\; 2\; \backslash cos\; x)\}\{a^2\; +\; 4\}\; +\; \backslash frac\{1.2\; e^\{ax\}\}\{a(a^2\; +\; 4)\}$

575. $\backslash int\; e^\{ax\}\; \backslash sin^3\; x\; dx\; =\; \backslash frac\{e^\{ax\}\; \backslash sin^2\; x\; (a\; \backslash sin\; x\; -\; 3\; \backslash cos\; x)\}\{a^2\; +\; 9\}\; +\; \backslash frac\{2.3\; e^\{ax\}\; (a\; \backslash sin\; x\; -\; \backslash cos\; x)\}\{(a^2\; +\; 1)(a^2\; +\; 9)\}$

1. 1. $dx$
2. 2. $adx$
3. 3. $x^n\; dx$
4. 4. $y\; dx$
5. 5. $y\; dz$
6. 6. $\text{'}dy$
7. 7. $x^m\; (a+bx)^\{-n\}\; dx$
8. 30. $x^m\; (a+bx^2)^\{-n\}\; dx$
9. 51. $x^m\; (a+bx+cx^2)^\{-n\}\; dx$
10. 72. $x^m\; (a+bx^3)^\{-n\}\; dx$
11. 90. $x^\{n-m\}\; (a+bx^n)^\{-1\}\; dx$
12. 92. $x^\{2n-m\}\; (a+bx^n+cx^\{2n\})^\{-1\}\; dx$
13. 93. $x^m\; ([x+f][x+g]\; \backslash dots\; [x^2+ax+b])^\{-1\}\; \backslash dots\; dx$
14. 99. $x^m\; \backslash frac\{A+Bx+Cx^2\; \backslash dots\}\{a+bx+cx^2\; \backslash dots\}\; dx$
15. 102. $x^m\; (a+bx)^\{\backslash frac\{n\}\{2\}\}\; dx$
16. 144. $x^m\; (a+bx)^\{\backslash frac\{n\}\{3\}\}\; dx$
17. 172. $x^m\; (a+bx^2)^\{\backslash frac\{n\}\{2\}\}\; dx$
18. 216. $x^m\; (ax+bx^2)^\{\backslash frac\{n\}\{2\}\}\; dx$
19. 258. $x^m\; (a+bx+cx^2)^\{\backslash frac\{n\}\{2\}\}\; dx$
20. 300. $x^\{\backslash frac\{m\}\{2\}\}\; (a+bx)^n\; dx$
21. 301. $x^\{m-\backslash frac\{1\}\{2\}\}\; (a+bx^2)^\{-n\}\; dx$