An Elementary Treatise On Curves, Functions, And Forces
An Elementary Treatise On Curves, Functions, And Forces
Benjamin Peirce
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Integrate — ^ . Ans. \os.^ x. ^ X log. X ^ 50. Problem. To find the value of the definite in- tegral n. (-\og. X)i (419) in which n is an integer greater than — 2. Solution. First. In this case, (408-410) give /. X=I, F. X^^x, f. F. Xz=x', (420) /. (- log. X. Y = X (- log. Xf + ^f. (- log. X)^-' (421) in which when x =1 0, or = ], (422) the first term of the second member vanishes as in example 2 of B. II. § 109, so that the required integral becomes In-i-^og-^-'- (423) By this process, then, t...he exponent of ( — log. X) is dimin- ished by unity ; and a continued repetition of it gives for the value of (419) i(i-')(i-"") (l-''+^)f:-i-^o, .. Mm n in which h is an integer not greater than - -f- 1. «5> 50. ] LOGARITHMIC FUNCTIONS. 105 Logarithmic definite integrals. Secondly. When n is even, let h — ^n (425) and (424) gives /i. (-Iog. 2;)'^=1. 2. 3 h. (426) Thirdly. When n is odd, and positive, let h = ln + i, (427) and (424) gives /■. (-log.^)'-*= (A_J)(A_|)... |. J/.. (_Iog..
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