Suppose first that x is positive then we shall write

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Suppose first that x is positive. Then we shall write Z dx x = log x, (2) and we shall call the function on the right-hand side of this equation the logarithmic function : it is defined so far only for positive values of x . Next suppose x negative. Then - x is positive, and so log( - x ) is defined by what precedes. Also d dx log( - x ) = - 1 - x = 1 x ,
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[VI : 129] DERIVATIVES AND INTEGRALS 280 so that, when x is negative, Z dx x = log( - x ) . (3) The formulae (2) and (3) may be united in the formulae Z dx x = log( ± x ) = log | x | , (4) where the ambiguous sign is to be chosen so that ± x is positive: these formulae hold for all real values of x other than x = 0. The most fundamental of the properties of log x which will be proved in Ch. IX are expressed by the equations log 1 = 0 , log(1 /x ) = - log x, log xy = log x + log y, of which the second is an obvious deduction from the first and third. It is not really necessary, for the purposes of this chapter, to assume the truth of any of these formulae; but they sometimes enable us to write our formulae in a more compact form than would otherwise be possible. It follows from the last of the formulae that log x 2 is equal to 2 log x if x > 0 and to 2 log( - x ) if x < 0, and in either case to 2 log | x | . Either of the formulae (4) is therefore equivalent to the formula Z dx x = 1 2 log x 2 . (5) The five formulae (1)–(3) are the five most fundamental standard forms of the Integral Calculus. To them should be added two more, viz. Z dx 1 + x 2 = arc tan x, Z x 1 - x 2 = ± arc sin x. * (6) * See § 119 for the rule for determining the ambiguous sign.
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[VI : 130] DERIVATIVES AND INTEGRALS 281 129. Polynomials. All the general theorems of § 113 may of course also be stated as theorems in integration. Thus we have, to begin with, the formulae Z { f ( x ) + F ( x ) } dx = Z f ( x ) dx + Z F ( x ) dx, (1) Z kf ( x ) dx = k Z f ( x ) dx. (2) Here it is assumed, of course, that the arbitrary constants are adjusted properly. Thus the formula (1) asserts that the sum of any integral of f ( x ) and any integral of F ( x ) is an integral of f ( x ) + F ( x ). These theorems enable us to write down at once the integral of any function of the form A ν f ν ( x ), the sum of a finite number of constant multiples of functions whose integrals are known. In particular we can write down the integral of any polynomial : thus Z ( a 0 x n + a 1 x n - 1 + · · · + a n ) dx = a 0 x n +1 n + 1 + a 1 x n n + · · · + a n x. 130. Rational Functions. After integrating polynomials it is natu- ral to turn our attention next to rational functions . Let us suppose R ( x ) to be any rational function expressed in the standard form of § 117 , viz. as the sum of a polynomial Π( x ) and a number of terms of the form A/ ( x - α ) p . We can at once write down the integrals of the polynomial and of all the other terms except those for which p = 1, since Z A ( x - α ) p dx = - A p - 1 1 ( x - α ) p - 1 , whether α be real or complex ( § 117 ).
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