Leybourn's Math. Rep. vol. i. 62. Integration by Infinite Series. p. 74. If the proposed fluent is not integrable by substitution, as in Art. 29, nor by either of the methods of continuation and of indeterminate coefficients, and if it cannot be reduced to a known form by any artifice of calculation, it may always be expanded into an infinite series, each term of which may be integrated. The developement of a function into a series is also of great importance in other branches of the subject. The principal object, in the developement, is to obtain series which will converge with the greatest rapidity. If a large integral value is to be assigned to the variable, we shall require a descending series; if it is to be a small fraction, an ascending series; and in either case we may obtain an approximate value of the fluent. It may be observed in this place, that when the series does not terminate, the symbol, which connects the function and its developement, does not necessarily represent numerical equality; for if the series does not converge, it is the symbol of a quantity of which the mind can form no precise idea, and even if it does converge, it may only represent an approximate value. In these cases, all that the symbol denotes, is, that the two functions are convertible into each other by division, evolution, or some other analytical process. Vid. Woodhouse's "Principles of Analytical Calculation," p. 3. = 63. Transcendental Fluents. These are, in general, to be integrated by the same methods as the algebraick. By transformation they may always be converted into algebraick fluents; but it frequently requires great analytical skill to reduce them to elementary forms. When the fluent is partly algebraick and partly transcendental, integration by parts will almost always be found to be the method which should be adopted. In all cases, they, as well as algebraick fluents, may be expanded into series, and an approximate value may be obtained. 64. Exponential and Logarithmick Fluents. It is evident that the form fada, where n is a positive integer, may be reduced and integrated by the same method. Assume Q1a* .. dq=—dB+ladA .'. B= la + a laa la2 adx 2x2 2 25 x Hence it is manifest that the is an integer, may be reduced to -Q+la. A, form fa'da, where n adr Ꮳ an elementary form which has hitherto baffled the skill of the most eminent analysts, and which can be integrated only by develope dc-x-2a da Assume q=x¬1a.. dq=-dc-ladB ..-c=Q-la. B. But a similar assumption will not give These four examples might have been integrated by substituting y = a*. Substitute y=x.. y=x2 and 2y3dy=xdx.. du=2e"y3dy .. (Ex. 1.) u = 2e" {y3-3y2 + 2.3y-2.3} 3 = 2e11 { x2 - 3x+2.3x2 -2.3}. Ex. 7. dulx2. x3dx. dx Assume Pla2.x-2.. dr=-2du+2x-2lx. |