A Series of Mathematical Texts For Colleges Edited by EARLE RAYMOND HEDRICK TABLES OF INTEGRALS AND OT,HER MATHEMATICAL DATA THE MACMILLAN COMPANY NEW YORK . CHICAGO DALLAS. dTI,ANTh . BAN FRANCISCO LONDON . M*NILA BRETT-MACMILLAN LTD. TORONTO f i i TABLES OF INTEGRALS AND OTHER MATHEMATICAL DATA HERBERT BRISTOL DWIGHT, D.Sc. Professor of Electrical Machinery Massachusetts Insta’tute of Technology THIRD EDITION .New York THE MACMILLAN COMPANY Third Edition @ The Macmillan Company 1957 All rights reserved-no part of this book may be reproduced in any form without permission in writing from the publisher, except by a reviewer who wishes to quote brief passages in connection with a review written for inclusion in magazine or newspaper. Printed in the United States of America Firsl Printing Previous editions copyright, 1934, 1947, by The Marmillan Company Library of Congress catalog card number: 57-7909 PREFACE TO THE FIRST EDITION The first study of any portion of mathematics should not be done from a synopsis of compact results, such as this collection. The references, although they are far from complete, will be helpful, it is hoped, in showing where the derivation of the results is given or where further similar results may be found. A list of numbered references is given at the end of the book. These are referred to in the text as “Ref. 7, p. 32,” etc., the page num- ber being that of the publication to which reference is made. Letters are considered to represent real quantities unless other- wise stated. Where the square root of a quantity is indicated, the positive value is to be taken, unless otherwise indicated. Two vertical lines enclosing a quantity represent the absolute or numerical value of that quantity, that is, the modulus of the quantity. The absolute value is a positive quantity. Thus, log I- 31 = log 3. The constant of integration is to be understood after each integral. The integrals may usually be checked by differentiat- ing. In algebraic expressions, the symbol log represents natural or Napierian logarithms, that is, logarithms to the base e. When any other base is intended, it will be indicated in the usual .~ manner. When an integral contains the logarithm of a certain quantity, integration should not be carried from a negative to a positive value of that quantity. If the quantity is negative, the logarithm of the absolute value of the quantity may be used, since log (- 1) = (2k + 1) ?ri will be part of the constant of integration (see 409.03). Accordingly, in many cases, the loga- rithm of an absolute value is shown, in giving an integral, so as to indicate that it applies to real values, both positive and negative. Inverse trigonometric functions are to be understood as refer- ring to the principal values. Suggestions and criticisms as to the material of this book and as to errors that may be in it, will be welcomed. v vi PREFACE The author desires to acknowledge valuable suggestions from Professors P. Franklin, W. H. Timbie, L. F. Woodruff, and F. S. Woods, of Massachusetts Institute of Technology. H. B. DWIGHT. CAMBRIDGE, MASS. December, 1933. PREFACE TO THE SECOND EDITION A considerable number of items have been added, including groups of integrals involving (ax2 + 62~ + fP2, r+kx and a + l cos 2 ) also additional material on inverse functions of complex quanti- ties and on Bessel functions. A probability integral table (No. 1045) has been included. It is desired to express appreciation for valuable suggestions from Professor Wm. R. Smythe of California Institute of Tech- nology and for the continued help and interest of Professor Philip Franklin of the Department of Mathematics, Massachusetts In- stitute of Technology. HERBERT B. DWIGHT. CAMBRIDGE, MASS. PREFACE TO THE THIRD EDITION In this edition, items 59.1 and 59.2 on determinants have been added. The group (No. 512) of derivatives of inverse trigo- nometric functions has been made more complete. On page 271 material is given, suggested by Dr. Rose M. Ring, which extends the tables of ez and e-z considerably, and is convenient when a calculating machine is used. Tables 1015 and 1016 of trigonometric functions of hundredths of degrees are given in this edition on pages 220 to 257. When calculating machines are used, the angles of a problem are PREFACE vii usually given in decimals. A great many trigonometric formulas involve addition of angles or multiplication of them by some quantity, and even when the angles are given in degrees, minutes, and seconds, to change the values to decimals of a degree gives the advantages that are always afforded by a decimal system compared with older and more awkward units. In such cases, the tables in hundredths of degrees are advantageous. HERBERT B. DWIGHT LEXINGTON, MASS. 38 RATIONAL ALGEBRAIC FUNCTIONS Integrals Involving Crrf x4 170. - dX x2 + ax42 + a2 s a4 + x4 x2 - ax42 + a2 1 ax42 -0 + 275&j$j tJan-l a2 _ x2 x dx 170.1. - = s a4 + x4 x2dx x2 + ax42 + a2 170.2. - = - &kx2 s a4 + x4 - ax42 + a2 1 -. ax42 + -2 tan-’ ($2 - x2 x3dx 170.3. - = a log (a” + x4), s a4 + x4 x3dx 171.3. - = - alog Ia4 - x41. s a4 - x4 dx 173. = -&og Zm . s x(a + bxm) I a + bxm I CONTENTS ITEMwo. P&m 1. ALGEBRAIC FUNCTIONS. ............................... 1 60. Algebraic Functions-Derivatives. ..................... 14 80. Rational Algebraic Functions-Integrals. ............... 16 180. Irrational Algebraic Functions-Integrals. ............... 39 400. TRIGONOMETRIC FUNCTIONS. ........................... 73 427. Trigonometric Functions-Derivatives. ................. 87 429. Trigonometric Functions-Integrals. ................... 87 500. INVERSE TRIGONOMETRIC FUNCTIONS. ................... 112 512. Inverse Trigonometric Functions-Derivatives. .......... 115 515. Inverse Trigonometric Functions-Integrals. ............ 116 550. EXPONENTIAL FUNCTIONS. ............................. 125 563. Exponential Functions-Derivatives. ................... 126 565. Exponential Functions-Integrals. ..................... 126 585. PROBABILITY INTEGRALS. .............................. 129 600. LOGARITHMIC FUNCTIONS. ............................. 130 610. Logarithmic Functions-Integrals. ..................... 133 650. HYPERBOLIC FUNCTIONS. .............................. 143 667. Hyperbolic Functions-Derivatives. .................... 146 670. Hyperbolic Functions-Integrals. ...................... 147 700. INVERSE HYPERBOLIC FUNCTIONS. ...................... 156 728. Inverse Hyperbolic Functions-Derivatives. ............. 160 730. Inverse Hyperbolic Functions-Integrals. ............... 160 750. ELLIPTIC FUNCTIONS. ......................... 168 768. Elliptic Functions-Derivatives. ............... 170 770. Elliptic Functions-Integrals. ................. 170 800. BESSEL FUNCTIONS. .......................... 174 835. Bessel Functions-Integrals. ................... 191 840. SURFACE ZONAL HARMONICS. .................. 192 850. DEFINITE INTEGRALS. ........................ 194 890. DIFFERENTIAL EQUATIONS. ............................ 204 ix CONTENTS APPENDIX PAGE A. TABLES OF NUMERICAL VALUES ............................ 209 TABLE 1000. Values of du2 + b2/a ............................. 210 1005. Gamma Function ................................. 212 1010. Trigonometric Functions (Degrees and Minutes). .... 213 1011. Degrees, Minutes, and Seconds to Radians. ......... 218 1012. Radians to Degrees, Minutes, and Seconds........... 219 1015. Trigonometric Functions: Sin and Cos of Hundredths of Degrees ....................................... 220 1016. Trigonometric Functions: Tan and Cot of Hundredths of Degrees ....................................... 238 1020. Logarithms to Base 10. ........................... 258 1025. Natural Logarithms ............................... 260 1030. Exponential and Hyperbolic Functions .............. 264 1040. Complete Elliptic Integrals of the First Kind. ....... 272 1041. Complete Elliptic Integrals of the Second Kind. ..... 274 1045. Normal Probability Integral. ...................... 275 1050. Bessel Functions. ................................. 276 1060. Some Numerical Constants. ....................... 283 1070. Greek Alphabet .................................. 283 B. REFERENCES ............................................. 284 INDEX..................................................... 287 TABLES OF INTEGRALS AND OTHER MATHEMATICAL DATA . TABLES OF INTEGRALS AND OTHER MATHEMATICAL DATA ALGEBRAIC FUNCTIONS n(n - l)(n - 2) ti I. (1 + a$” = 1 + nx +@2, - llx2 + 31 . nl + + (n - qrtX' + -*** Note that, here and elsewhere, we take O! = 1. If n is a positive integer, the expression consists of a finite number of terms. If n is not a positive integer, the series is convergent for 1” < 1; and if n > 0, the series is con- vergent also for x2 = 1. mef. 21, p. 88.1 2. The coefficient of z+ in No. 1 is denoted by : or ,C,. 0 Values are given in the following table. TABLE OF BINOMIAL COEFFICIENTS ,C;: Values of n in left column; values of T in top row oll 2 3 4 5 ===,=r=zwm=-----===,=Yzzzwm=-----1 +p+p“i’p-pq i :1 ii N.B. Sum of any two adja- i 1 cent numbers in sameame row is 4 : 2 6 equal to number just below 5 1;: i the right-hand one of them. x :: 20 t 7t : ;fl 21ii 3556 l%itz 126:A ii:: t3 1 10 1 10 45 1% 2io 252 210 13260 4: 1: 1 For a large table see Ref. 59, v. 1, second section, p. 69. n(n -- 1) n(n - l)(n - 2) ~ 3. (1 - x)* = 1 - nx + -2r x2 - 3! + . + (-- l)r(n f&!” + -**- [See Table 2 and note under No. 1.1 4. (a&x)“=al(l*gn. 1 2 ALGEBRAIC FUNCTIONS 4.2. (1 f z)~ = 1 f 2x + x2. , 4.3. (1 f x)3 = 1 f 3x + 3x2 f 2. 4.4. (1 f x)” = 1 f 4x + 6x2 f 4d + 9, and so forth, using coefficients from Table 2. 5.1. [x2 2 11. 5.2. [x” 5 11. 5.3. (1&4”2 y *ix -LAx2* !A29 2-4-6 [x2 5 11. 5.4. (1fx)~‘~=1*;x+~x+$-6za + g#?+ =F ;..&.!.gy&+ + - * *, [x2 s 11. 5.5. (1 *zy2 = 1 *;x + [x2 5 11. 6. (1 + gp = 1 - nx + “‘n2T 1) $2 _ n(n + 13)r(n + 2) $3 (n+r-l)! . + *-- + (- 1)’ @, _ l)ir! xr + -**> [x2 < 11. 7. (1 _ x)-?a = 1 + nx + n(n2t 1) ,&2 + n(n + ‘,‘1” + 2) $3 . +(n+T’-l)!xr+ . + (72 - l)! r! [x2 < 1-J. 8. (a f x)d = u-n (1 f y, [x2 < a2]. SERIES AND FORMULAS 3 1.5.9.13 tiT . [z” < 11. + 4.8.12.16 1.4.7$ 9.02. (l&z)““=lr;x+~PT- 3.6.9 cs < a [a9 < 11. [a+ < 11. [a+ < 11. 9.06. (1 f z)+ =1=1=2~+3a+~2@+5ti=i=.*., [39 < 13.