To find P, Q.---S, which are constants, use I. 11 (b). 4. Third Case. f(x) not expressible as a product of real simple factors. Avoid imaginaries by using quadratic factors; suppose them all different. E.g. f(x)=a(x2 + kx + 1)(x2 + mx +n)(x − p)(x − q) - F(x) Kx+L Mx+N = + ƒ(x) = x2+kx+1* x2+mx+n + Р + Determine the constants K, L, M, etc., by I. 11. (b). Or see Williamson, § 1. 5. Fourth Case. Some quadratic factors repeated. f(x)=a(x2+kx+1)3 (x2 + mx + n)(x -p)---- F(x)_ K1x+L1 + + E.g. Determine the constants K1 L1 etc., by I. 11. (b), or as in Williamson, § 43. II. CHIEF METHODS OF TRANSFORMATION. 2. Alternative method. Use X=e.. 2 cos x = X + X-1 2i sin x = X-X-1 2 cos nx= X" + X-". 2i sin nx= X" - X-". Express sines and cosines of x or its multiples in terms of X. Multiply out. Collect pairs of terms of the form C (X"+X-"), and reintroduce sines and cosines. E.g. Sin3x cos 2x cos x TABLE OF INTEGRALS, III. A. 1-10. III.-IX.-TABLE OF INTEGRALS. a2 ax + b 7. fxTM(ax+b)"dæ. Use II. A., or IX. A. 1, or III. A. 20. 8. 1 dx =C+tan-1x = C - cot-1x. 1+x 9. C++ log [«<1] 1- x2 -X (-응), 1 = dx X2 (See III. A. 8.) ~ ~√(−ab) / 1-X: where X=z √/(-;), when ab<0. (See III. A. 9.) 11. = C+ log. (Ax+B ax + b xmdx 12. 1+xn (-1)m, = C + ( − 1)" log.(1 + x) - 1 Σ { cos1(m + 1)" log,(22 – 2x cos n n (Ax+B)(ax+b) Ab-aB where m is a positive integer or 0, and n a positive integer. If mn, use II. E. 1; if m<n, thus: Υπ n +1)} Υπ sin n Σsin (m+1)π N.B.-If n is odd, r takes the values 1, 3, 5----(n-2). where m is a positive integer or 0, and n is an odd positive integer. =(-1)m+1 xdx xn [XmdX where X=-x. (See III. A. 12.) where m is a positive integer or 0, and n is an even positive integer. See III. A. 12, if m and n are positive integers. See III. A. 13, 14, if m and n are positive integers. Otherwise, see IX. A. I. |