40.] Integration of (2 ax-x2) dx, and of (2 ax + x2)3 dx. [(2 ax−x2)* dx = [ {a2—(x− a)2}1 d(x− a) ; f{a2 which latter integral, if x-a = z, becomes {a2—z2}1dz, and is therefore of the form (79); and we have [(2 ax+x2)1 dx = [{(x+a)2 − a3}1d(x+a) = = =√ a2 2 log {x+a+ (2 ax+x2)1}. (83) = dx (a + bx + cx2) Hence = (+1) 1 2 a2 = k2; 1 dx 612 4ac-b2) + 4c2 {(x + x c 2 dx = dz; whence we have (84) SECTION 4.-Integration of Irrational Functions by 43.] Many infinitesimal elements involving irrational quantities may by a judicious substitution be transformed into equivalent integral and rational functions, and consequently integrated by the methods which have been investigated in the first two sections of the present chapter; the process of such transformation is called Rationalization, and we proceed to inquire into the conditions requisite for its application in certain cases. p To find the integral of xTM (a+bx") dx, where m, n, p, q are constants, integral or fractional, positive or negative. 929-1 +(-a) nb" 1 m + 1 is an integer, (za—a) is of a rational 1-n dz; m+1 1 (z — a) dz. (86) nb n m+1 -1 n n panded by the Binomial theorem, and each term of it having been multiplied by zp+q-1 may be integrated by means of Art. 11. -1 is negative, the integration may be accomplished by means of Section 2, and chiefly by the Reductionformulæ of Arts. 27-30. 44.] Again, as the element-function (a+bx")dx may be written in the form and as this is the same as that of equation (86), it follows that by substituting b+ax¬" = 29, shall be able to integrate by known methods. + n is an integer, by substituting a + bx” = za ; is an integer, by substituting b + ax¬"=z? ̧* * For other methods of Rationalization, and indeed for a complete collection of integrals of all kinds, the reader is referred to "Sammlung von Integraltafeln," von Ferdinand Minding; Berlin, 1849. Let a2+x2 = z2; .. x = (z2 — a2)3; dx = z dz 1, and is integral. q = 22; x2 dx = = = 3a2 23 203 = 3a2 (a2+x2) dx = 46.] Integration of = {2, (a+b), (a+b), dr R ex where R is the symbol of a rational function. Let be the least common multiple of the denominators of the fractional indices; and let us assume where R(2) denotes a rational function of z. |