38 CHAPTER III. FORMULE OF REDUCTION. 30. LET a + bx" be denoted by X; by integration by parts we have The equation (1) is called a formula of reduction; by means of it we make the integral of 1X depend on that of x+n-1 X-1. In the same way the latter integral can be made to depend on that of +2n-1 XP-2; and thus, if p be an integer we may proceed until we arrive at +1 XPP, that is +1, which is immediately integrable. m xm-n Xp+1 m x XP m m-1 XP dx. Ρ into p+1; thus m n bn (p+1) bn (p + 1) This formula may be used when we wish to make the integral of "X" depend upon another in which the exponent of x is diminished and that of X increased. For example, if m = 3, n = 2, and p=-2, we have The latter integral has already been determined, and thus the proposed integration is accomplished. m-1 Since JaXdx = fax (a + bx") dx Change m into m―n and transpose, then In (2), change m into m+n and p into p − 1, then dx m+n-1 1 Also for "X"dx = a faXde + b fax + xr+ dx, Change p into p+1 and transpose; thus 31. If an example is proposed to which one of the preceding formulæ is applicable, we may either quote that particular formula or may obtain the required result inde pendently. Thus, suppose we require xm-1 dx √(c2-x2) ; we have −√√ (c2 — x2) xm−2 + (m−2) c2 This result agrees with the equation (4) of the preceding article if we make a = c2, b = −1, n=2, p = xdx Another example is furnished by √(2) be written xdx which may ; if in equation (4) of the preceding − 1, n = 1, p = into 2a and m+respectively, we have xm dx √(2αx-x2) -1, and change a and m 32. In equation (6) of Art. 30 put a = c2, m = 1, n = 2, b=1, and p-r; thus '+ c2)r ̄ ̄ 2 (r − 1) c2 (x2 + c2)TM1 † 2 (r−1) c2 √) (x2 + c2)r-1 ° which occurs in Art. 18; for this last expression may be written thus 33. These formulæ of reduction are most useful when the integral has to be taken between certain limits. (x), x(x), (x), functions of x, such that as is obvious from Art. 3. For example, it may be shewn that n Suppose n 2 suppose positive quantity, then x (c2x2) vanishes both when x=0 and when x = c. Hence The following is a similar example. By integration by Thus if r be an integer we may reduce the integral to 34. The integration of trigonometrical functions is facilitated by formulæ of reduction. Let (sin x, cos x) denote any function of sin x and cos x; then if we put sin x = z, we have dz [$ (sin a, cos x) da = fø (z, √/(1 − 2)} de de = f$ {z, √(1 − dz For example, let & (sin x, cos x) = sin x cos2 x; then If in the six formulæ of Art. 30 we put a =1, b=-1, n=2, p =(q-1), we have |