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when i = 0; which quantity is determinate in form, but undetermined in value, because y and v are arbitrary constants.

91.] The values of the definite integrals given in the two preceding Articles are indeterminate because they involve the arbitrary constants j and v; and they are consequently called general definite integrals : there is however no sufficient reason why u and v should not be equal, and why indeed u and v should not each be equal to unity. If u=v=l, then the integral assumes a determinate form, and has been called by Cauchy the principal value of the definite integral. Thus for the principal value of the definite integral given in (42), we have

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value is 0; since if u = v= 1,1
Also when a is less than 6, for the general value of

pa dx

Ho a + b cos x we have, by (49),

{"_dx - 1

Jo a + b cos X (62 - a) and as the right-hand member = 0, when u = v= 1, the principal value of the integral = 0. Thus then,

s dx

Jo a +b cos x - TOP _22171 = 0, = 0, according as a is greater than, equal to, or less than b. Hence it appears that (a? 62)} /* _dx :

T h at bcosa is a discontinuous function of a and b; it is equal to 1 for all values of a greater than b; it is equal to oo, when a=b, and it is equal to 0 for all values of a less than b; and thus it abruptly changes its value from a constant quantity to zero, by passing through oo , when a becomes equal to b. Suppose this definite integral to represent the ordinate of a locus of which a is the abscissa; the locus will be a straight line, parallel to the x-axis, at a distance, = 1, from it, for all values of the abscissa greater than b; and for all values of the abscissa less than 6, it will be a straight line coincident with the axis of r; and the ordinate is infinite when the abscissa = b. Thus a definite integral may be a new species of transcendent, and may express a discontinuous function; indeed it often does so; in the sequel many cases will occur, and we shall have occasion to exhibit this particular characteristic in some very striking and important forms.

92.] Indefinite integrals often take a form which is incompatible with the remarks made at the end of Art. 83: although the element-function has the same sign throughout the range of integration, yet the indefinite integral has a contrary sign. Thus for instance, (cotx) dx = ) {(cosec x)2 – 1} dx = – cotx — X ;

(54) that is, the sum of a series of positive quantities is a negative quantity : which it of course cannot be. The preceding theory removes the apparent contradiction. Let the limits, whatever they may be, be introduced, and let the integrals be definite; then the result is correct. Thus, ${cot aye dx = [-cot x = x]

4

=14+1 = 4*; (55) which is a positive result. Also again,

in dx

Ju x2 m – 2m-1 lx. 2m-1 + 2mwhich is an absurd result; because the right-hand member is negative, although all the element-functions of which it is the sum are positive. But x^2m is infinite, when x = 0, 0 being a value included within the range of integration. We must therefore divide the integral into two parts, and we will take the principal value of each : thus,

(55)

(56)

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(57)

which is equal to co , when i = 0; and this is the correct value of the integral.

SECTION 3.—The Transformation of Definite Integrals by a

. Change of Variable. 93.] The general theory of the transformation of a single definite integral by means of a change of variable, and the effects of the substitution of a new variable, have been explained in Theorem IV, Art 8: and various examples of substitution have already been made in Chapters II and III, wherein indefinite integrals have been investigated. Further applications of the theory are now required, and with respect to a change of the limits of integration as well as to that of the element-function ; the latter part of the theory alone having been required in the preceding Chapters.

The equivalence of the following expressions is evident on inspection.

(58)

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(59)

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(60)

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If it is required to transform *F'(x) dx into an equivalent definite integral of which the limits are zn and zo, both zn and z. being finite quantities, the following substitution may be made.

(61)

Let

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The following are instances of this formula.

Suppose that the limits of the transformed integral are 1 and 0: then, 2n = 1, 2, = 0; and

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the variable 2 having been replaced by x in the right-hand

member by reason of (58). This is the formula of transformation from the old limits X, and X, to new limits 1 and 0.

Other cases of the preceding formula are the following: let zn = xn—&o, and zo = 0: then I F'() dx = | F'(x+x.) dx.

(65) Let zn = Xo, zo = Xn, so that the limits may be reversed; then

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The formula of transformation given in (61) is inapplicable when either zn or zo = o. If however a transformation is required so that the new limits may be oo and 0; then let

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(71)

20 The preceding are the formulæ of transformation which are generally applicable for the change of limits. Other formulæ will be applied in the sequel, for the special problem will usually suggest the required substitution.

94.] The following are examples of transformation in particular integrals.

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=p; and for z substitute x in the last integral;

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Again in this last integral, let x = (tan 6)2; then

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consequently adding this to the former value of u,

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Thus the definite integral is evaluated for the given limits without a previous knowledge of the indefinite integral. Many similar instances will occur hereafter; and indeed the investigation of these values is a capital part of our Treatise.

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