differential expressions to known forms, and when this cannot be effected, the expression may be expanded in a series of which the several terms can be integrated by elementary forms. 82. Since a constant multiplier or divisor in a function is retained in the differential (11), it follows that a constant multiplier or divisor may be removed and placed before the sign of integration thus fax" dx x dx 1 = a fx" dx, and fx" dx. In the differential calculus it S α = a was shown (11) that the differential of the sum of any number of functions is equal to the sum of the differentials of those functions; hence to integrate an expression consisting of any number of differentials connected by the signs plus or minus, we must take the integral of cach separately, and connect them by their proper signs: thus f(ax" dx-bx dx +c x2 dx) = a fx" d x − b fx" d x + c f x1 d x + C ; where one constant quantity is annexed, because the aggregate of three or of any number of constants is only one constant. 83. It will be convenient to collect the various elementary forms of integrals, derived from reversing the fundamental processes in the differential calculus, and place them in a tabular form for reference. Hence if In a similar manner from the inverse trigonometrical functions we get the following elementary forms: Si S dx √ (a2 - x2) dx a2 + x2 :- dx 1 α dx a2 + x2 == 84. It may be worthy of remark here that the same differential admits of two apparently different integral forms: thus in (16) we have + C, and also dx √ (a2 — x2) a but since sin-1 a = constant; therefore sin a + constant; which shows that the two expressions include precisely the same system of values, when all possible constant values are given to the arbitrary constants. The two integrals differ only by a constant quantity. In a similar manner the forms in (17), (18), and (19), may be shown to be identical; hence if the radius be unity, or a = 1, we have 85. The forms (16), (17), (18), (19), may be made more general in the following manner. Writing bx for x, and b d x for dx, we have (16) bdx Again in (19) write 262 a x instead of x, and dx instead of dx; In this manner we obtain the four following forms :— a2 dx 1 b √ (a2 — b2x2) a b =- COS x + C'.. (20). dx a2 + b2x2 x + C a b dx dx -1 a b cot-1 -x+C'.. (21). Cosec a b -1 α 262 -1 Covers x+C'(23). a2 b x + C' (22). 86. Since d (uv) = udv+vdu, where u and v are functions of x; therefore by integration, uv = fudv+fvdu, and transposing Sudv v = u v ― ƒ v du This is called the formula of integration by parts, and enables us to find the integral of u dv, provided the integral of vdu can be found. The method of integration by parts is extensively employed in reducing integrals to known forms. If we change u into u1or, then we have 1 (25), a formula which is sometimes advantageously employed. 87. We may now proceed to explain the methods of reducing integrals of different functions to one or other of the preceding fundamental forms; but before entering upon the various artifices to be employed in these transformations, we shall advert to one or two very useful theorems which are of frequent occurrence in the integral calculus. Thus since dx” = n x2-1 dx; therefore fx-'d x = +C; hence to integrate a differential in which the differential of a variable is multiplied by a power of that variable, the index of the power being constant, we have the following rule: 1 n Divide by the differential of the variable, add unity to the index ; divide by the index thus increased, and annex the constant quantity. Thus fx* dx = x2 + C, ƒ x* dx = ‡ x2 + C, fxdx fx dx = 2x++ C, f x* dx 3 This principle may be extended to the integration of the general expression (ax+b) xdx. For if we put a x" + b = z, then differentiating we have f(ax+b)" xdx= na z dz; therefore na (m+1) na (m + 1) The principle here employed is applicable in all cases when the index (n-1) of the power of x without the vinculum is less by unity than the index (n) of the power within it, except in the cases when 1, or n = 0, which reduce the differential expression to the integrals of which will be considered in the next Article. The substitution of z for ax+b was employed to reduce the d f ferential to the form (1), but in practice the principle should be applied directly to the differential whose integral is required. Thus, con sidering (ax+b) as a monomial, its differential is na xdx, and (ax+b)TM na Now dividing the proposed differential by this, gives increase the index m by unity, divide by m+1, the index thus increased, and we obtain (ax+b)TM+1 f(ax+b)" x1 d x = na (m + 1) + C. ; therefore, conversely = log (x + a) + C'; whence it follows that, when the numerator of a differential is the differential of the denominator, the integral is the napierian logarithm of the denominator. The constant may be incorporated with log (x + a), for the constant C' is the logarithm of another constant C, that is, C' = log C; hence = log (x + a) + log C = log C (x + a). dx x + a Reverting to the differential in Article 87, we have, when m = log C (ax+b). And when n = O, in the same differential, we have S(a+b)m dx x (a + b) = f da 89. We frequently meet with integrals which may be brought to elementary forms by certain easy algebraic processes, as well as by the substitution of certain differential forms for others equivalent to them, and to assist the student in the transformation and reduction of such integrals, we shall here give the following table. (10.) (11.) (12.) (l – x) = (1 − ) (1 - x)*= (1 - x)* _ (1−x)*. (1 − 2) = (*) (z− − 1) = 2* (*−1). = (13.) (a + bx')* = b+ (14.) (15.) (16.) 8 90. Many other equivalent expressions will suggest themselves in practice, especially in trigonometrical functions, and several of those now given will be employed in the examples immediately following, as well as in the integration of some differentials whose integrals involve logarithms, and in the process of integration by parts, both of which will presently come under consideration. |
