Also dX' Here is the x-gradient of X', and is called the "second dx differential coefficient of X with respect to x," or, more simply, the "second x-gradient of X." It is concisely written X". Using this notation (X") and applying § 126 we have Here the given function is f'(X), and the supposition is that it is directly integrable with respect to X, but not so with respect to x. On this supposition the transformation will be of use if it is found that f(x) .X" is directly, or more easily, integrable with respect to x. 129. General Reduction for X".-If ƒ(X) = X", then ƒ(X) Xr+1 r+1 ; so that in this case the above formula would be = SXdx, which In some cases this form may be preferable to dX, would be given by § 122. 130. General Reduction of "X".—If in § 126 one of the two functions whose product is to be integrated be xm and the other X", where m and r are any constant indices, the transformation gives JamXdx = xm+1Xr If this latter quantity be not directly integrable, it may still be capable of being further reduced by the application of other formulas of transformation already explained, so as to finally reduce it to a directly integrable form. Such a formula is the base of certain Formulas of Reduction. 131. Conditions of Utility of Same.-The last formula given is capable of repeated application, provided that X' is proportional either to some power of x or to some power of X, the right-hand integral then reducing to the same general form as the left-hand one. In either case, or again in the case xX'=a+bX, it is not difficult to prove that X must be of the form X= a + bx". If r be a positive integer, then X" can be expanded into a finite series of powers of x, which when multiplied by am will give another series of powers of x, each term of which can be integrated separately; so that in this case no need of the above reduction formula will arise; although in some cases its use may shorten the work involved. But the formula is useful for repeated reductions if is negative or fractional. Various cases of such uses are given in Section IX. of the Classified Reference List at the end. 132. Reduction of (a+ba")": Here we have fa"X'd.c on each side. Bringing these two terms to one side, and dividing out by the sum of their numerical factors, a formula of reduction by which in the integration the power of X is reduced by 1, while that of x is left unchanged. The reduction of the power of X is compensated for by the multiplication (outside the sign of integration) by the factor a, which has the same dimensions as X. 66 This formula can be used inversely to pass from 2Xˆ-1 to x2X2, that is, to increase the power of X by 1 without changing that of x. * See Classified List, IX. A. 1. If the other form of X', namely, nban-1, be used in this transformation, there results a formula of reduction by which, while the power of X is decreased by 1, that of x is increased by n. "+nX-1dx may be converted into a By the previous formula Sam+nxr-1 quantity in terms of fam+"X′dæ, and thus tegral in which the power of x is raised by n, while that of X is left unaltered. By similar transformations one can ring the changes among the integrals of the following set of nine functions, any one of which can be reduced to any other. The complete set of reduction formulæ for this purpose are given in Section IX. of the Classified Reference List at the end. 133. Reduction of th Power of Series of any Powers of x.— Similarly if The last integral may be taken in terms each of the form and on account of the reduction from r to (r-1) in the index of X, these may be more amenable to simple integration than the original famXdx. Evidently this formula applies to the sum of any number of terms in different powers of x. 134. Special Case.-If in the last article one index = O and another = 1, we have X= a + bx" + cx in which case a simple reduction, like that of § 132, will show that 1 xm+1Xr xm [ xmx dx = x + 1 x n + m + 1 + myr fx" {(n − 1)ex + na}X'-'dx. m+1+nr nr 135. Trigonometrical Reductions. If in the general formula of 126 the product XE' be equal to sin"x, then we may split this into the two factors sin x and sin"-1x, thus: X'E- (n - 1) sin^-2x cos2x = − (n − 1) sin”-2x(1 − sin 2x) = (n-1) sin"x-(n-1) sin"-2x. by adding together on the left-hand side the two terms in sinxdx and dividing by the sum of their coefficients, namely, 1+ (n − 1) = n . 136. Trigonometrical Reductions. Since by § 78 the anglegradient of the tangent is the reciprocal of the square of the cosine, dX and since dx = we have X and, therefore, dxcos2xd(tan x) = (1 − sin2x)d(tan x) [tan"-2xdx = [ ti tan"-2x(1-sin2x)d(tan x) = n - I By either the process adopted in this article or that of last article, the various trigonometrical formulæ of reduction are established which are set forth in order in Section IX. B. 1 to 12 of the Classified Reference List at the end of this book. In each case the reduction changes the index by 2, which change results from the substitution sin2 = 1 - cos2x. * See Classified List, IX. B. 1. If n the index to be reduced be an even integer, a repetition of the reduction will finally bring down the integral to the form If n be odd, the finally reduced integral fdx, whose integral is x. which have already been demonstrated. cos xdx or tan xdx, etc., etc., all of If in these 137. Trigonometrico-Algebraic Substitution. dX trigonometrical reductions n is fractional, then since dx= and d sin x dx = X' cos x, etc., etc., we may convert the formula into that of § 132 and IX. A. 1 of the Classified List, thus: 138. Composite Trigonometrical Reduction.-From the elementary formulæ for the sine and cosine of the sum of two angles, in terms of the sines and cosines of these angles, we may write : Since p (p-1) + 1 = sin px = sin (p-1) x. cos x + cos (p − 1) x sin x. cos px = cos (p-1)x. cos x - sin (p-1)x. sinx. Therefore integrals of (sin+x. sin px), and (sin±x. cos px), and (cost. sin px) and (cost. cos px) may, by repeated application of the above formulæ, be reduced to series of integrals of the form (sintx cost), provided p be an integer. For these latter forms, see Classified Reference List, IX. B. 7-10. |