[ ̃ˆ {x′(x) ±ƒ ̃′(x) ±$'(x) ± ...} dx = [x(x)±ƒ(x) ± $(x) ± ...]“ = F(X„)—F(x。)±{ƒ (xn)−−−ƒ (x0)}+{$(x„)−$(xo)} ±..... = [** ¥′(x) dx ± [**ƒ'(x)dx ± [* ¢'(x)dx ± = + The same theorem is also true of indefinite integrals. ... (19) ["{¥'(x) + √=1ƒ'(x)} dx = [*x(x)dx + √=1 [*ƒ′(x)dx. (20) dæ THEOREM III.-If the infinitesimal element is of the form f(x) × v′(x) dx, then = [**ƒ (x) ×¥'(x) dx = [f(x) × F(x) ]* − [**F(x) ׃′(x) dx. Το For convenience of notation, let ƒ(x) = u, F (x) = v, F′(x) dx dv, and let up, U1, Uq... Un, Vo, V1, V2, ... V n be the several values of u and v corresponding to xo, x1, x2,... x, respectively; then udv = u(v1− v。) +U1 (V2−V1) +Uq (V3 —V2) + . . . +Un−1(Vn—Vn−1) n = u„v„—u。v。— {v1 (U1 −uo) +v2(U2—U2) + ... +ν„(U„—un-1)} ; and since the differences between vo, v1, v2, ... v, are infinitesimal, as are also the differences between u。, u1, if we take i to be the general symbol of an infinitesimal, we have Un ù dv = u„v„—u。。— {v% (U1−uo) +V1 (U2−U1) + ... + V„−1(Un—Un_1)} — {i̟ ̧(u1—u。)+i1⁄2 (U2−u1) + ... + in(Un—Un−1)} ; the last term of which equality must be neglected, because it contains infinitesimals of a higher order than those of the pre ceding term, and the preceding term is [**vdu; therefore This theorem is known by the name of integration by parts, and is of very frequent use; the form which it assumes in the case of an indefinite integral is THEOREM IV.-If, in order to determine the Integral of F'(x) dx, it is convenient to introduce another variable z, related to a by the equation z = (x), so that x= f(z), dx = f'(z) dz, F′(x) = F′(f(z)) ; and if z, and z。 are the values of z corresponding to x, and xo, that is, for the definite integral determined by means of x, we may take as its equivalent the other definite integral determined by means of z. Let Z1, Z2, ... Zn-1 Z1 be the values of z corresponding to x1, x2, ...1; then, the elements of a being infinitesimal, we have x1-xo = f(z1)-ƒ (≈0) = ƒ'(≈0) (≈1—Zo), Xn—Xn_1 = ƒ (Zn) —ƒ (Zn−1) = ƒ' (Zn−1) (Zn — Zn−1). F'(x) dx = (X1−X。)F′(x)+(X2−X1) F′ (X1) + . . . + (xn−X n−1) F′ (xn−1) = = F(ƒ (≈0))ƒ'(≈0)(≈1 — 20) + F′ (ƒ (≈1)) ƒ' (≈1) (Zg—≈1) + ..... that is, the two definite integrals are equal; and the latter therefore may be used for the former; and vice versa. This method is called Integration by substitution, and is of course true for the indefinite integrals as far as variable quantities enter into the functions; the arbitrary constants will however frequently assume different although equivalent forms. Other Theorems on definite integrals we shall reserve to Chapter IV. 9.] The determination of definite integrals from first principles. xn dx. Let, as in Art. 4, x-xo be divided into n infinitesimal parts, and let x1, x2,...,- be the values of a corresponding to the points of partition; then dx = x1-xo+X2−X1+X3−X 2 + ... + X-X-1 be the values of a corre Let, as heretofore, x1, X2, X3, ..... Xn-1 sponding to the (n-1) points of partition of x-xo; and as the mode of partition does not affect the result provided that the elements are infinitesimal, let us suppose that xo, x1, ... x, form a geometrical progression whose common ratio differs infinitesimally from unity: that is, let for if i is infinitesimal, (1+i)+1—1 = (a+1)i. |