And thus fxdxx2± a2 may be reduced in the same manner as xdx provided that p is a positive integer. The results A, B, C, &c., and A', B', c', &c. show that when the index p is odd, the fluent may be obtained in Algebraick terms; but that when it is even, the integration depends upon the quadrature of the circle or of the hyperbola. When P is odd the forms are integrable by the method of Art. 46. 56. All fluents of the forms s xdx and fxdx √x2±2ax, where p is a positive integer, may be reduced to the above forms. For substitute y = x ± a, and the forms are reduced to of (y±a)'dy and ƒ(y± a)3 √y2±a2 dy, which, since p is a positive integer, will terminate when expanded, and each term may be integrated as in the preceding articles. 57. Integral Tables. Meyer Hirsch, a German mathematician, has published a collection of Tables, in which are registered those Integral Formula which occur the most frequently in calculation. This work, which has been lately translated, is doubtless a useful book of reference, but yet the student should not rest satisfied with a knowledge of the methods of Integration, but should be able himself to integrate the Formulæ without referring to the tables. In these tables, at pages 119 and 130 (English translaxdx tion) S = 1 to p and fxdx xa are calculated from 11 or 12; and it is obvious that if all of the first form had been alone calculated, they would have enabled us to integrate the latter form, which is the same as xdx x2+2dx ± a2 S 58. The process of reducing du to the forms... dc, dв, da is, in general, rendered more simple by assuming certain rectangles... P, Q, R, Such that de may contain du and dc, do may contain dc and dв, and dr may contain dв and da. The rules for the assumption of the rectangles will be given in the second volume; but we shall show the method by applying it to the forms of Articles 54 and 55, when p is a negative integer. Ex. 1. Required S dx either positive or negative. where a and b may be dx x3 √ a+bx2 The value of в might have been obtained from Art. 13. Ex. 2. Required f = C. known transcendental form. Ex. 3. Required ƒ x ̄ˆdx √a+ bx2 = c. Let da dx va+bx2 dв=x-2dx\/a+bx2 Assume p =x−3. (a + bx2)3. 3dx(a + bx2)2 dp= + 3bdв =-3adc .. c= (a+bx2) where p is an integer, either positive or negative, may always be reduced and integrated. It may be observed, that by these assumptions p is increased or diminished each time by 2. By proper assumptions, in all cases of a binomial fluent, the index of a without the vinculum may be increased or diminished by its index under the vinculum. Ex. 5. u=f dx Assume Px.(x2+1)-2.. dp= dx 4x2dx Ꮖ (x2 + 1)2 ̄ ̄ (x2+1)3 4dx (x2+1)3 (x2 + 1)3 dx-3x dx -3(x2+1)dx = + (x2+1) 3 P Assume = x(x2 + 1)−1.•. dq= |