8. 23 dx = 2·3026 - - - - log10X + C = log. X+C 10. APPROXIMATE INTEGRATION. -a SXdz=" (i.) a 2n -n-1 =a+ n 2(b− a) =a+ b =a+r. a { Xo + X2n + 4(X1 + Xg + Xs+----) 3 f(x)dx=h| } { ƒ(b) +ƒ(a)} +f(a+h) +f(a+2h) 11. METHOD OF UNDETERMINED COEFFICIENTS. (i.) If a function is known to be expressible in a certain form containing unknown coefficients, these coefficients can be determined by transforming the identity (see below) and equating coefficients of like terms on either side; like terms being those whose variable part is the same function of the variable (e.g., the same power, or the same trigonometrical function), or in both of which alike the variable is absent (constant terms). The transformation referred to may be : (a) Differentiating both sides (as in III. B. 18). (b) Clearing of fractions (as in II. E. 2, etc.), or otherwise. (ii.) Another method: Give to the primitive variable as many different values, in the identity, as there are coefficients to be determined; whereby we get as many equations. as are necessary to determine them. GENERAL NOTE re IMAGINARIES. Some of the formulas here given contain parts which would become imaginary if the quantities involved took values outside certain limits: becoming, e.g., square roots and logarithms of negative quantities, inverse sines of quantities greater than unity, etc. When a definite integral is deduced from the indefinite one, these imaginaries, explicitly or implicitly, cancel one another, if the subject of integration is itself real. But, in many instances, two or more forms are given for the integral of the same function (e.g., III. B. 6, VI. 5, 6), of which that one is to be selected which, for the values of the constants in the particular problem under consideration, is free from imaginaries. Some of these formulas contain parts which are imaginary for certain values of x only, whatever the constants may be, and others do so for all values of x when the constants are outside certain limits. E.g., the formula sin-1 is imaginary when x>a, x α a and + a. On the other hand, the contains imaginaries when b is negative, whatever the value of x may be. In these classified tables, the conditions under which a formula involves imaginaries are, as a rule, pointed out in cases of the latter sort, but not in those of the former. I. GENERAL THEOREMS. 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. A. II.-CHIEF METHODS OF TRANSFORMATION. 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.) B. C. E.g., [log{(1+2x)(1 +3x)}dæ= [{log(1 + 2x) + log(1 + 3x)}dæ =(log(1 + 2x)dx + flog(1 + 3x)dx. Add and subtract the same quantity. E.g., 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 II. CHIEF METHODS OF TRANSFORMATION G-E. G. Substitute ƒ(X) for x and ƒ'(X)dX for dx. (See I. 6.) Sp. Case: ƒ(X)=pX+q= x } . dx = pdX 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. where m and n are positive integers. 1. If mn reduce F(x) + H +h by ordinary division to an integral function of x,+ a fraction of similar form to the above, in which f(x) m<n. 2. First Case. f(x)=a product of simple factors, all different. f(x)=a(x − p)(x − q) - - - - (x − s) F(x) Р Q S = + -+ x-q |