SECTION 4.-Integration of Irrational Functions by Rationalization. 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. To find the integral of rm (a + bxcm)ī dx, where m, n, p, q are constants, integral or fractional, positive or negative. 1 (27-ain Let a + bx;" = %?; form : and if m :-1 is positive, (29– a) " may be expanded by the Binomial theorem, and each term of it having been multiplied by zp+q-1 may be integrated by means of Art. 11. And if + -1 is negative, the integration may be accomplished by means of Section 2, and chiefly by the Reductionformula of Arts. 27–30. 44.] Again, as the element-function wm(a + b**)7 dx may be written in the form c***(6+ ax-»)dx ; (87) and as this is the same as that of equation (86), it follows that by substituting b+ax-" = 29, the result will be rational if " + + 2 is an integer; and we na shall be able to integrate by known methods. Hence we may by means of rationalization determine f1z(a + bamáda, (1) when" is an integer, by substituting a + bx * = zo; (2) when“ +? is an integer, by substituting b+ ax-*=27.* ng 45.] Examples of the two preceding articles. * For other methods of Rationalization, and indeed for a complete colleetion of integrals of all kinds, the reader is referred to “Sammlung von Integraltafeln," von Ferdinand Minding; Berlin, 1849. BY RATIONALIZATION hon 46.] Integration a + bx19 c+ex!? Ic+ex! where R is the symbol of a rational function. Let I be the least common multiple of the denominators of the fractional indices; and let us assume a + bx cm - a = z; ii. x = 1(bc-ae)zl-1 C + ex dz. (6-ez'j2 a+bx Platbariz r Also c+ ex) = z?, ( ctenl = 2', ......; all of which are rational; and therefore dim tatbrigatori 1c + ex! ' c+ ex where R,(z) denotes a rational function of z. |