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 New Online Book! Handbook of Mathematical Functions (AMS55) Conversion & Calculation Home >> Reference Information Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (AMS55) Purchase the electronic edition of this book in Adobe PDF format! FIRST | PREVIOUS | NEXT | LAST | CONTENTS | PAGE | ABOUT | SMALL | MEDIUM | LARGE Below is the OCR-scanned text from this page: 12 ELEMENTARY ANALYTICAL METHODS Leibniz's Theorem for Differentiation of a Product 3.3.8 d" d"u n d"-'udv n d"-2ud2v ds" ( u v ) = a v+ (1) d5n-' da: ' ( 2 ) d2n-2 d22 3.3.9 3.3.10 Integration by Parts 3.3.12 s u d v = uv - s v d u 3.3.13 Suvdx=(sudx) v - s ( s u d x ) 2 dx Integrals of Rational Algebraic Functions (Integration cons tan ts are omitted) 3.3.14 J(ax+b)"dx= (ax+b)"+' a(n+l) (nz-l) 3.3.15 The following formulas are useful for evaluating s ( u x ~ ~ ~ ~ c ) n where P ( x ) is a polynomial and n>l is aninteger. 3.3.16 2ax+ b (4m- b y arctan a x 2 s (a?+ bx+c) = (4m - b2) * (b2-4m<0) - 3.3.17 ( b2-4ac>0) -2 2ax+b =- ( b2-4m=0) 3.3.18 3.3.19 1 --In Iax2+bx+cI-- 3.3.20 In - c+dx (adZbC) dx 1 d x 1 ,f(a+bx) ( c + d x ] = a lu+bJ 3.3.21 3.3.22 3.3.23 X X arctan-+ d x dx -X 3.3024 s(x2+,2)2=~ a 2a2(x2+a2) Integrals of Irrational Algebraic Functions dx - 2 arctan [ - d ( a + b x ) ] 2 (baO, dO) d(u+bx) ' I 2 2 -- 3.3.28 arctan [ ] (d(ad-bc)