42 INFINITESIMAL CALCULUS /: / >" + / y.r+i/rte in which /u, v are new positive arbitrary constants. In all cases, the general values of the definite integrals /x /-+< /(xXfa, / f(x}dx , J cc deduced from the preceding equations, depend on the form of the function /(.), and may be finite and determinate, or infinite, or in determinate, depending on the values attributed to the arbitrary constants fj., v, /j. lt v it . . . fj.,,, v n . Whenever the integrals become indeterminate, if each of the con stants /*, v, . . . /u,,, V H , be made unity, the corresponding values of f(x}dx and f(z)djs become lim. and lim. f(x)dx~ _i ...+/ f(x)dx~] - .rn-f-e J These are called, by Cauchy, the principal values of the definite integrals /*X /"+oo / f(x)dx and / f(x}dx , Jl f, J - <X> in the case in question. Again, the definite integral if f(x) be finite when x = a, is infinitely small if the difference between the limits a and b is an evanescent quantity. But, iff(x) become infinitely great at the same time, the value of the definite integral may be finite, or even infinite. In the latter cases the integral is called a singular definite integral. For instance, if /(a^) = oo , the integral f l f(x)dx, where is an infinitesimal, is of this class. Its value may be represented by the method of 137 ; for, if /j denote the limit of (v x 1 )f(x 1 ) when x = x lt we have / Jxtl f(x}dx=f l log Again, if the limits a and b each become infinite, while preserving the same sign, we have another class of singular definite integrals, such as f in which e is considered evanescent as before. In this, as in the former case, if xj[x} tend to a limiting value/, when x is infinitely great, we shall have 141. We sliall illustrate the preceding by a few simple examples commencing with the definite integral -* Here the function becomes infinite when x=0, and we have
but fe X - = log , -l" dr. f^d =/
- > J.r a ~
, pt = log f Accordingly, the principal value of / - is log ( " v ) , and its general value is log + log ^ . The latter is perfectly arbitrary a o v and indeterminate. Again, each of the singular definite integrals is equal to log . v , /-* (lr ,. f f-He d.r f X ( 7, r Next / ^=11111. / ;,:>+/ -> /-*<> -.r ;/ " y^e j ~ But /-*"* l.l,/ X ^.l 1; y_j. ar /xe y- y ya a; 2 ^e X fx-dx ri i i in .-. / -f-lim. + -- -v- y_ a . o a;- 1 LM " X a J Consequently, the principal value and the general value of the de finite integral are botli infinite in this case. In like manner 11 1 Accordingly, the general value of the integral is infinite, while its principal value is % ( - - J . x A / Next let us consider the singular definite integral r~vf (x a}dx If we substitute the proposed limits, and afterwards make e = 0, we readily find the value of the proposed to be log ( ) an inde- V " / terminate quantity, as v is by hypothesis supposed to be arbitrary. Likewise + 00 Accordingly the general value of (x - a)dx A , when ,-0. is perfectly arbitrary, while its principal value is zero. In like manner, since f dx __ 1 , -ifx-a we find the general and also the principal value of / +c dx _ZT J_^ (x-a)* + b*~b- Again, it readily follows from the last result that, when aoi 2 , the value of the definite integral /~ x dx ir / -I- 9Av -L /.-> 1S 00 142. Next let us consider the definite interal w , dx . Here But Also yo = lim. e t(^) ^ = /" ^ e .f W^ , making ax - e . T / yz/e where e is infinitely small.
