(n − 1) (tan ¿)^-1 − fdæ (tan x)=2 (tan æ)3 – tan æ + x. (tan æ)ˆ − 4 (tan æ)* + 1⁄2 (tan x)2 + log (cos x). = -(cota)+(cot x) + log (sin x). (18) If the function be " cos x, the formula of reduction is fdx x cos x = a" sin x + næ2-1 cos x fdx x2 cos x = x2 sin x + 2x cos x fdx x3 cos x = x23 sin x + 3æ2 cos x In the same way we find fdx x sin x = -x cos x + sin x. 2-2 n (n − 1) fdx x2-2 cos x. 2 sin x. 6x sin x 6 cos x. fdæ æsina-x'cosa + 4a3sina +12a2 cosa - 24a sina - 24 cosø. (19) If the function be e" (cos a)" the formula of reduction is (2n-3) a (n-2) + dx (n−1)(a2-b3) (a+b cos x)-1 (n−1)(a2-b2)√ (a+bcos x)" -2 √ Let n = 2, then dx (a + b cos x)2 CHAPTER III. INTEGRATION OF DIFFERENTIAL FUNCTIONS OF TWO OR MORE VARIABLES. SECT. 1. Functions of the first order. IN order that a differential function of two variables of the first order, such as Pdx + Qdy, should be the differential of a function u, it is necessary that the condition The application of these formulæ may be generally facilitated by observing that in the second term of the former it is only necessary to integrate the terms in Q which involve only, and in the latter those terms of P which involve y only. therefore d P = d Q 0 = Integrating with respect to y, u = by2 + fdx {a + (1 + 2)}}; Integrating with respect to y, and observing that there is no term in P involving y only, we find Since P does not contain any term independent of x, whence x + (x2 + y2)} y + C'; U = = log C {x + (x2 + y3)}}. (4) Let (a3y + x3) dx + (b3 + a2x) dy The integral of this is = du. (5) Let (3xy x2) dx − (1 + 6y2 - 3x2y) dy = du; 1 then dP = dQ 6xy= |