38 INFINITESIMAL CALCULUS reducible to Eulerian integrals, functions of which a short discus sion will be subsequently given. The following examples are given for the purpose of illustrating the preceding results. /- 1 Ex. 1. / (1 - x*) m dx , where m is an integer. JQ , 2.4.6... Am. (2m) Ex. 2. f 2 Jo siaxdx . 3.5.7... (2w + l) 3 . 6 . 12 Ans. 5 . 11 . 17 r " dx 3. / 7-3 ~s
where n is an integer.
Ans _ TT 1^3. 5. .. (2-3) E.r. 4 . / a cos"+ 2r yo .
a3 cos nx dx . (n + 2r)(n + 2r-l). . - (n + r+l) 2 +- -+i 1.2.3 ... r .". 5. To deduce "Vallis s value for IT "by aid of the definite integrals considered in this article. When m is positive, we have, for all values of % between and, 2* S i n 2m-] a;> S in 2m .> sin.- m + l x ; accordingly, 7T I 2 y a sin 2m #<:fa> I 2
2m lies between 2 - 4 " 2 > 3. 5 . 7 ... (2m- 1) and 2.4.6... 2m 3. 5. 7 ... (2m + 1) but when m is indefinitely increased the latter fractions tend to equality, and, consequently, we have the well-known formula of Wallis, viz. : ie ,..,,,2.2 4.4 6.6 = hmit of __._.____... 123. As a further example of the method of successive reduction, we shall consider the integral fx Here, integrating by parts, we have !Lr_l f x (- x - n -dx. m J n -^dx -fx m -l - x) n - ] dx . l fv m -l-x)-*dx, -lj Again, fx m (l - 8)- 3 d-/a? m - 1 (l - Substituting, and transposing, we get /V-i(l-iE)- 1 fte- !g " (1 " a;)l " 1 + J m+n-1 m+n By successive applications of this relation the proposed integral can be found whenever n is a positive integer. It can be determined in like manner when in is a positive integer. The integral of x m (a + bx)"dx readily admits of similar treatment. The preceding is a simple case of the integration of what arc styled JL binomial differentials, i.e., differentials of the form x m (a + bx n )i dx, in which m, n, p, q represent any numbers, positive or negative. We propose to determine in what cases such differentials can be immediately integrated by a transformation. Assume a + bx n = zi, then x = l n , and we get p+q-l The latter can be immedately integrated when is an integej. Again, substituting for x, the expression x m (a + lx n ) q dx be- J> o P. comes -z m n i ~ (az n + b] q dz . This can be integrated whenever p , - f is an integer. n q It can be shown that when neither of these conditions is fulfilled the integral of the binomial differential cannot be expressed except by infinite series. 124. Irrational Functions. We shall next briefly consider the method of proceeding in the case of irrational expressions. Suppose F(x, /X)dx to represent the expression whose integration is sought, where F is a rational algebraic function, and X is a rational integer polynomial of any degree in x. Here, since even powers of VX are rational, and odd powers contain VX as a factor, it is plain that F(x, VX) can bo always reduced to the form P + QVx P + Q VX where P, Q, 1", Q are rational algebraic functions of x. Again, if this bo multiplied by P Q VX, it is reducible to the form M + N VX", or to M + -^L , Vx where M and N" are rational functions. Consequently integrals of the proposed form are reducible in general to two parts, of which one is rational, and the other is of the form /(*) rf(x) ^dx_ J 000 VT It can be shown that, when X contains powers of x beyond the second, such integrals cannot be reduced to any of the elementary forms given at the commencement; and, accordingly, they depend on higher transcendental functions. When X is a cubic or a bi quadratic, such integrals are reducible to elliptic functions, of which a short account shall be given below. When X is a polynomial of higher degree, the integrals are usually styled hyper-elliptic integrals. They were first treated of in a general manner by Abel. 125. We shall at present consider only the case where X is a qua dratic, of the form a + 2bx + cx~. The integral f(x) dx f can be rendered rational in different ways. (1) First, let the roots of a + 2bx + ex 2 = be real, and suppose a + 2bx + ex 2 = c(x - a) (x - /3). fl~2 If c be positive, we assume x a = (x j3)-: 2 , or x = - --- ; then , and dx = - V. dz 1-z 2 Hence the transformed expression is a rational function of z. _l_ /3-2 If c be negative, we make x= , aud the transformed ex pression is rational, in like manner. When the roots a and /8 are imaginary this method of trans formation introduces imaginary forms into our results. In such cases it is usually more advantageous to adopt a di tit-rent treat ment. For instance, if we assume we get Hence and n + 2bx = ~ 2 - 2.rc dx V a + 2bx + ex* b + zVc This substitution consequently furnishes a rational function in z. Again, when c is negative the expression becomes rational by the assumption /a + xz . a + 2bx In general, if we substitute >/ f , for x, where , , ,u, 2 , A + 2fiz + j I/, i>, i/ satisfy the equations /u 2 - v = a, v + Kv 2/u/x = 2b, /t 2 - A.V = c, it can be shown without difficulty that dx 2dz . aud accordingly the function f(x) dx becomes rational by this transformation. This last is a particular case of the general method adopted by Jacobi (Fundamenta nova thcorife functionum ellipticarum] for the transformation of elliptic integrals. 126. The class of integrals here discussed admits also of another mode of treatment.
