40.] Integration of (2 ax-x2) dx, and of (2 ax + x2)1 dx. [(2 ax−x2)1 dx = [ {aa—(x − a)3}+d(x−a) ; which latter integral, if a—a = z, becomes ({a2 — 22}1dz, and is therefore of the form (79); and we have (2 f{2ax+x2)1 dx = {{(x+a)3 — a2} *d(x+a) 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. To find the integral of xTM (a+bx")" dæ, where m, n, p, q are constants, integral or fractional, positive or negative. 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. And if m+1 n -1 is negative, the integration may be accomplished by means of Section 2, and chiefly by the Reductionformulæ of Arts. 27-30. P 44.] Again, as the element-function (a+bx") dx may be written in the form m+ пр p X a (b+ax-")dx; (87) and as this is the same as that of equation (86), it follows that by shall be able to integrate by known methods. n SxTM (a + bxTM‚¶dx, is an integer, by substituting a+bx" = 2; (1) when m+1 *For other methods of Rationalization, and indeed for a complete collec tion of integrals of all kinds, the reader is referred to "Sammlung von Inte. graltafeln," von Ferdinand Minding; Berlin, 1849. |