. An elementary course of infinitesimal calculus . j> (aj) dx,Ja Ja rb rb-a I (l>{x)dx= I 0 (a! + a) dx,Ja Ja rb \ rkb I (kx) cfe = T / {x)dx (1). J a Evidently, 7 is a function of the limits of integration a, b,and will in general vary when either of these varies. 220 INFINITESIMAL CALCULUS. [CH. VI Regarding a as fixed, let us form the derived function of /with respect to the upper limit h. We have rb+Sb I + BI= ^{x)div J a /b rb+Sb)dx (2), by Art 89, 2°. Hence /•M-Si S/ = J (x)dx = ;{h + ehh) (3), by Art. 89, 3°. This shews that S7 vanishes with Sb, sothat / is a continuous fun
. An elementary course of infinitesimal calculus . j> (aj) dx,Ja Ja rb rb-a I (l>{x)dx= I 0 (a! + a) dx,Ja Ja rb \ rkb I (kx) cfe = T / {x)dx (1). J a Evidently, 7 is a function of the limits of integration a, b,and will in general vary when either of these varies. 220 INFINITESIMAL CALCULUS. [CH. VI Regarding a as fixed, let us form the derived function of /with respect to the upper limit h. We have rb+Sb I + BI= ^{x)div J a /b rb+Sb)dx (2), by Art 89, 2°. Hence /•M-Si S/ = J (x)dx = ;{h + ehh) (3), by Art. 89, 3°. This shews that S7 vanishes with Sb, sothat / is a continuous function of 6. Also, since g = ,/,(6 + 0S6) (4). we have, on proceeding to the limit (86 = 0),. Fig. 53. In the figure, OA = a, OB = 6, Bff = Sb, and 87 is representedby the rectangle having BB as its base. 90-92] DEFINITE INTEGRALS. 221 In the same way, if we regard the upper limit h as fixed,and the lower limit a as variable, we find that / is acontinuous function of a, and that J^-^^^) (^- 91. Existence of an Indefinite Integral. We can now shew that any function <f) (x), having thecharacter postulated in Art. 87, has an indefinite integral, there exists a definable (but not necessarily calculable)function yfr^ai) such that ylr{x) = <f,(x) (1), or f(a;)=l)-{a!) (2). For if we write ^(^)=j(x)dx (3), J a the expression on the right hand is, by Art. 87, a determinatefunction of ^, and the investigation just given shews that itsatisfies the condition t(^) = «^(a (4). The lower limit of integration in (3) is, from the presentpoint of view, arbitrary, and the function -yfr (f) is thereforeindeterminate to the extent of an additive constant. For,by Art.
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