SPECIAL NOTE AS TO LOGARITHMIC TERMS. When a term of the form A log X, where A is a constant, occurs in an integral, it becomes imaginary when x has such values as make X negative; but in such cases A log(-X), which is real, may always be used instead of A log X, since it has the same differential coefficient as A log X. This note applies to III. A. 3, 5, 11, 16, etc. II.-CHIEF METHODS OF TRANSFORMATION. A. Express the subject of integration as the sum of a series of terms, and integrate these separately (see I. 4). (Integration by decomposition or separation.) E.g., [log{(1+2x)(1 + 3x)}dæ= [ {log(1 +2x) + log(1 + 3x)}dæ =(log(1 + 2x)dx + (log(1 + 3x)dx. B. Add and subtract the same quantity. E.g., C. Multiply (or divide) numerator and denominator by the same quantity. E.g., E. Resolve rational fractions into partial fractions. (See p. 146.) F. Express a product of powers of sines and cosines as a sum of terms, each consisting of a sine or cosine multiplied by a constant. (See p. 147.) K G. Substitute f(X) for x and f'(X)dX for dr. (See I. 6.) In the case of a definite integral, change the limits correspondingly (See I. 7), or else transform back to x after integration, before assigning limits. (See p. 148.) H. Differentiate or integrate an integral with respect to any quantity in it which is not a function of x, and a new integral is deduced. K. Use integration by parts. (See I. 7.) II. D. Chief Methods of Expansion in Series : 1. Binomial Theorem. 2. Exponential Theorem. 3. Expansion of Log, (1±x). 4. Trigonometric series derived from the preceding expansions, by use of imaginaries. 5. Taylor's Theorem or Maclaurin's Theorem (including 1, 2, 3, as special cases). 6. Fourier's Expansion in series of sines and cosines. 7. Spherical Harmonics, Lamé's Functions, Bessel's Functions. Toroidal Functions, etc. II. E. Partial Fractions. F(x)_Ax+BxTM-1+----+ H m-1 where m and n are positive integers. 1. If mn reduce by ordinary division to an integral F(x) f(x) function of x,+ a fraction of similar form to the above, in which m<n. 2. First Case. f(x)=a product of simple factors, all different. f(x)=a(x-p)(x−q)----(x-8) To find P, Q.-S, which are constants, use I. 11 (b). P: Otherwise, F(p) F(q) 3. Second Case. Some factors repeated. E.g. f(x)=a(x− p)3(x − q)2(x − r) - - - F(x) P1 Po 1. Q2 1 2 P 3 Q1+ f(x)=x-p+ (x -p)2 + (x − p)s + x-q+ (x − q)2 + R + x-r To find P1 P2----use I. 11. (b) or as in Williamson, § 40. 2 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) - - - - = + Р + + Determine the constants K, L, M, etc., by I. 11. (b). Or see Williamson, § 1. 5. Fourth Case. Some quadratic factors repeated. E.g. + f(x) = x2+kx+1(x2+hx+1)2 + (x2 + kx+1)3 F(x) K1x+L1 K2x+L2 Kq+Lg + Determine the constants K1 L1 etc., by I. 11. (b), or as in Williamson, § 43. |