Baixe Preface The pur-pose of this handbook is to supply a collection of e outras Notas de estudo em PDF para Atualidades, somente na Docsity! EA DO LA” Second Edition A Rg RR DU a The perfect aid for better grades! e. —-— Half a million copies sold of the first edition! Es — More than 2400 formulas and tables e Covers elementary to advanced math Use with these courses: [7 College Mathemátios [57/Numerical Analysis [2 Galeulus [4 Caleulus Il [57 Galculus 1 [24 Differential Equations [34 Probability and Statistics P r e f a c e The pur-pose of this handbook is to supply a collection of mathematical formulas and tables which will prove to be valuable to students and research workers in the fields of mathematics, physics, engineering and other sciences. TO accomplish this, tare has been taken to include those formulas and tables which are most likely to be needed in practice rather than highly specialized results which are rarely used. Every effort has been made to present results concisely as well as precisely SO that they may be referred to with a maxi- mum of ease as well as confidence. Topics covered range from elementary to advanced. Elementary topics include those from algebra, geometry, trigonometry, analytic geometry and calculus. Advanced topics include those from differential equations, vector analysis, Fourier series, gamma and beta functions, Bessel and Legendre functions, Fourier and Laplace transforms, elliptic functions and various other special functions of importance. This wide coverage of topics has been adopted SO as to provide within a single volume most of the important mathematical results needed by the student or research worker regardless of his particular field of interest or level of attainment. The book is divided into two main parts. Part 1 presents mathematical formulas together with other material, such as definitions, theorems, graphs, diagrams, etc., essential for proper understanding and application of the formulas. Included in this first part are extensive tables of integrals and Laplace transforms which should be extremely useful to the student and research worker. Part II presents numerical tables such as the values of elementary functions (trigonometric, logarithmic, exponential, hyperbolic, etc.) as well as advanced functions (Bessel, Legendre, elliptic, etc.). In order to eliminate confusion, especially to the beginner in mathematics, the numerical tables for each function are sep- arated, Thus, for example, the sine and cosine functions for angles in degrees and minutes are given in separate tables rather than in one table SO that there is no need to be concerned about the possibility of errer due to looking in the wrong column or row. 1 wish to thank the various authors and publishers who gave me permission to adapt data from their books for use in several tables of this handbook. Appropriate references to such sources are given next to the corresponding tables. In particular 1 am indebted to the Literary Executor of the late Sir Ronald A. Fisher, F.R.S., to Dr. Frank Yates, F.R.S., and to Oliver and Boyd Ltd., Edinburgh, for permission to use data from Table III of their book S t a t i s t i c a l T a b l e s f o y B i o l o g i c a l , A g r i c u l t u r a l a n d M e d i c a l R e s e a r c h . 1 also wish to express my gratitude to Nicola Menti, Henry Hayden and Jack Margolin for their excellent editorial cooperation. M. R. SPIEGEL Rensselaer Polytechnic Institute September, 1968 CONTENTS Page 31. Hypergeometrie Functions ........cciiisiiis 32. Laplace Transforms.........icccs 33. Fourier Transforms........ciiiiisiiiiiiiii is 174 34. Elliptic Functions ............... 35. Miscellaneous Special Functions . 36. Inequalities .........cccciico 37. Partial Fraction Expansions.......ciicciic 187 38. infinite Products .......... 188 39. Probability Distributions . 189 40. Special Moments of Inertia.. 190 41. Conversion Factors.........ccliiciiisiiiisss ria . 192 Part H Sample problems illustrating use of the tables..........ccciiiiiiiiiciiiitesiieitrea 194 1. Four Place Common Logarithms ........ciic 202 2. Four Place Common Antilogarithms.. 3. Sing (x in degrees and minutes) ..... 4. Cosz (x in degrees and minutes) ..............oiilo 5. Tana (x in degrees and minutes) .......iicciiiiiiiiii eee ieerer cares 208 6. Cota (x in degrees and minutes)... 7. Secex (x in degrees and minutes) . 8. Cscx (x in degrees and minutes) ... 9. Natural Trigonometric Functions (in radians).......cccil iii i essere, 212 10. logsina (x in degrees and minutes) .. W. logcosz (x in degrees and minutes) 12. logtang (x in degrees and minutes)... 13. Conversion of radians to degrees, minutes and seconds or fractions of a degree .......... 14. Conversion of degrees, minutes and seconds to radians 15. Natural or Napierian Logarithms logex or ng.....cciciiiiiiiiiie 224 16. Exponential functions e” .... 17. Exponential functions e”. 184. Hyperbolic functions sinhz ... 18b. Hyperbolic functions coshg ........cciiiieii 230 18c. Hyperbolic functions tanhg........cccciiiiii A 282 CONTENTS Page 19. Faetorial mn... 234 20. Gamma Function... «284 21. Binomial Coeffcients .......ciiiiiiii 236 22. Squares, Cubes, Roots and Reciprocals 288 23. Compound Amount: (Lar... 240 24. Present Value of an Amount: (LAP... 241 25. Amount of an Annuity: Qro PPP PPP 242 26. Present Value of an Annuity: lo (rm 27. Bessel functions Jo(x)........ciiis iii 28. Bessel functions Ji(x).....cii 29. Bessel functions Po(x) ......eci . 30. Bessel functions Yi(x) ......ii 245 31. Bessel functions Jo(x) . . 246 32. Bessel Tunctions A(z)... 246 33. Bessel functions Ko(2) ......cciic iii «247 34. Bessel functions Ki(x) ......iiiciiii 247 35. Bessel functions Ber (x). ..248 36. Bessel functions Bei (x)... 248 37. Bessel functions Ker(x)........iiiiiiiis . 249 38. Bessel functions Kei (x) .....iiiiiiii 249 39. Values for Approximate Zeros of Bessel Functions.........lciiiiiiiiiiiio 250 40. Exponential, Sine and Cosine Integrals.........cciiiiiiiiiiii 251 41, Legendre Polynomials Pa(x) ............ ..252 42. Legendre Polynomials P,(cos 6) .......ciiriii 253 43. Complete Elliptic Integrals of First and Second Kinds ........ciiciiiiiiiiiio 25 44. Incomplete Elliptie Integral of the First Kind..........ciciiciiiiiiiiiiiiiiiiis 255 45. Incomplete Elliptic Integral of the Second Kind .........iiciiiiiiiiiae 255 46. Ordinates of the Standard Normal Curve..........iccciciisiesiiieri 256 47. Areas under the Standard Normal Curve..........ciiiiiiiiisiisiiiiiiiies 257 43. Percentile Values for Students t Distribution ...........ciciiiiisi iii 258 49. Percentile Values for the Chi Square Distribution .. ...259 50. 95th Percentile Values for the FP Distribution .. .260 51. 99th Percentile Values for the F Distribution .... -.261 52. Random Numbers.........cccciseiisisiiies isa ciis acena een 262 Index ot Special Symbols and Notations........iiiiiiciiiiiiiiii A 263 Index... sis seen ter acer aar eee 265 Part I FORMULAS nora assica sia natatosteritit Asses esa tá die E seo cieses sa santistatieo cast d tapar ant ri aas dna ri mano 2.1 2.2 23 24 2.5 2.6 27 2.8 2.9 2.10 teto? = (r—y? (e +a3 tr— a te+yt (em (e +98 (a —9)ê tetos (e) = The results 2.1 21 2.12 2.13 2.14 215 2.16 217 2.18 219 ay = 3-3 = +93 = ni-yi = 595 = +95 = es = n2+220y +32 “2 — Say dy? 28 + Saly + Jay + gp 23 — Buy + Boy? — yyê 24 + daty + Gutyl + 4my3 + qt nt — duty + Guy — duo + gy? 28 + Baty + 10342 + 10223 + Suyl + yê 28 — Baty + 10u8y? — 102298 + Sayt — y5 28 + Gxby + 15a4y2 + 200348 + 15a2ys + 6xyS + yê 28 — GuSy + L5xty? — 200848 + 150248 — 6xy5 + 9º to 2.10 above are special cases of the binomial formula [see page 3]. (x — ua + 9) (a — (a + cy + 92 (e + aa? — my +?) (a — gta + a)(a? + 992) (e Mat+ aty + all + ey + 48) (o aos — ay + a? — ayê +98 (e (a + (a? + ay + (aê — ay de 7) drprA = (rat ay +) at + dyt (82 + 2o0y + 22/02 — Day + 2y?) Some generalizations of the above are given by the following results where x is a positive integer. 2.20 2.21 2.22 2.23 anti — gên+ gnt + gone ato — yo er + tn (e — (ela + adnty + an 2yo 4 eo + gem) + A) 2m 4 -yla — e ya — + (x o(s 2xy cos + 4 Ya Zey cos 2nm (ue — rr pp (s Cy cos] + 9 ) e gts) dr Bla Ly pre + 2ay cosg o + *) (a + y)(nêm — glnc ty + atn-2y2 — = E 2 — teta + 2oy cosgã 2mz «fu + +) (= 2xy cos ty ) = (els (Bro A at2y + amy ce an 1 — any + an Sgt — 000) (e — ya + 9) (o — Bmy cost + va — 2xy cos à + *) 1) Zig Br 2 cos ET a Z Xe + Buy cos Y ) (2n — Dr fg? + fm lr 2 (= 2%y cos da + 4 ) . (o: — 2my costtDr a = (a + 2wy cos 2 PERIIS SIS PESE SSI REI SA PO IODO SIS a SRP TE SOS The BINOMIAL FOR: PesiResosiRo Sie dE, IH n=1,2,8,... factorialn or n factorial is defined as 31 al = ledegea a We also define zero factorial as 32 ot=1 H n=1,2,8,... then (nr) 21 =1a—2 33 LPS ed nam + nao 4 eia Dump + age This is called the binomial formula. It can be extended to other values of x and then is an infinite series [see Binomial Series, page 110! MpRtagsezaai o The result 3.3 can also be written ' 34 ra o as (O eectga (oct e (Deca 4 co + (Oo where the coefficients, called binomial coeficients, are given by PV nem B en k+I nt - n 35 (5) = k! > amo! O (124) 4 THE BINOMIAL FORMULA AND BINOMIAL COElFI?ICIFJNTS PROPERTIES OF BINOMIAL COEFFiClEblTS 3.6 This leads to Paseal’s triangk [sec page 2361. 3.7 (1) + (y) + (;) + ... + (1) = 27l 3.8 (1) - (y) + (;) - ..+-w(;) = 0 3.9 3.10 (;) + (;) + (7) + .*. = 2n-1 3.11 (y) + (;) + (i) + ..* = 2n-1 3.12 3.13 -d 3.14 q+n2+ ... +np = 72.. MUlTlNOMlAk FORfvlUlA 3.16 (zI+%~+...+zp)~ = ~~~!~~~~~..~~!~~1~~2...~~~ where the mm, denoted by 2, is taken over a11 nonnegative integers % %, . . , np fox- whkh G E O M E T R I C F O R M U L A S 7 4 . 1 7 A r e a = & n r 2 s i n s = 3 6 0 ° + n r 2 s i n n 4 . 1 8 P e r i m e t e r = 2 n r s i n z = 2 n r s i n y Fig. 4-10 4 . 1 9 A r e a = n r 2 t a n Z T = n r 2 t a n L ! T ! ! ? n n I T 4 . 2 0 P e r i m e t e r = 2 n r t a n k = 2 n r t a n ? 0 : F i g . 4 - 1 1 SRdMMHW W C%Ct& OF RADWS T 4 . 2 1 A r e a o f s h a d e d p a r t = + r 2 ( e - s i n e) e T r tz!? Fig. 4-12 4 . 2 2 4 . 2 3 A r e a = r a b 5 7r/2 P e r i m e t e r = 4a 4 1 - kz s i + e c l @ 0 = 27r@sTq [ a p p r o x i m a t e l y ] w h e r e k = ~/=/a. See p a g e 254 f o r n u m e r i c a l t a b l e s . F i g . 4 - 1 3 4 . 2 4 A r e a = $ab 4 . 2 5 A r c l e n g t h ABC = -& dw + E l n 4 a + @ T T G 1 ) AOC b Fig. 4-14 f - 8 GEOMETRIC FORMULAS RECTANGULAR PARALLELEPIPED OF LENGTH u, HEIGHT r?, WIDTH c 4.26 Volume = ubc 4.27 Surface area = Z(ab + CLC + bc) PARALLELEPIPED OF CROSS-SECTIONAL AREA A AND HEIGHT h 4.28 Volume = Ah = abcsine 4.29 4.30 4.31 4.32 4.33 4.34 a Fig. 4-15 Fig. 4-16 SPHERE OF RADIUS ,r Volume = + Surface area = 4wz 1 ,------- ---x . @ Fig. 4-17 RIGHT CIRCULAR CYLINDER OF RADIUS T AND HEIGHT h Volume = 77&2 Lateral surface area = 25dz h Fig. 4-18 CIRCULAR CYLINDER OF RADIUS r AND SLANT HEIGHT 2 Volume = m2h = ~41 sine 2wh Lateral surface area = 2777-1 = z = 2wh csc e Fig. 4-19 GEOMETRIC FORMULAS 9 CYLINDER OF CROSS-SECTIONAL AREA A AND SLANT HEIGHT I 4.35 Volume = Ah = Alsine 4.36 Ph - Lateral surface area = pZ = G - ph csc t Note that formulas 4.31 to 4.34 are special cases. Fig. 4-20 RIGHT CIRCULAR CONE OF RADIUS ,r AND HEIGHT h 4.37 Volume = jîw2/z 4.38 Lateral surface area = 77rd77-D = ~-7-1 Fig. 4-21 PYRAMID OF BASE AREA A AND HEIGHT h 4.39 Volume = +Ah Fig. 4-22 SPHERICAL CAP OF RADIUS ,r AND HEIGHT h 4.40 Volume (shaded in figure) = &rIt2(3v - h) 4.41 Surface area = 2wh Fig. 4-23 FRUSTRUM OF RIGHT CIRCULAR CONE OF RADII u,h AND HEIGHT h 4.42 Volume = +h(d + ab + b2) 4.43 Lateral surface area = T(U + b) dF + (b - CL)~ = n(a+b)l Fig. 4-24 12 TRIGONOMETRIC FUNCTIONS For an angle A in any quadrant the trigonometric functions of A are defined as follows. 5.7 sin A = ylr 5.8 COS A = xl?. 5.9 tan A = ylx 5.10 cet A = xly 5.11 sec A = v-lx 5.12 csc A = riy RELAT!ONSHiP BETWEEN DEGREES AN0 RAnIANS A radian is that angle e subtended at tenter 0 of a eircle by an arc MN equal to the radius r. Since 2~ radians = 360° we have 5.13 1 radian = 180°/~ = 57.29577 95130 8232. . . o 5.14 10 = ~/180 radians = 0.01745 32925 19943 29576 92.. .radians N 1 r e B 0 r M Fig. 5-4 REkATlONSHlPS AMONG TRtGONOMETRK FUNCTItB4S 5.15 tanA = 5 5.19 sine A + ~OS~A = 1 5.16 1 COS A &A ~II ~ zz - tan A sin A 5.20 sec2A - tane A = 1 5.17 1 sec A = ~ COS A 5.21 csce A - cots A = 1 5.18 1 cscA = - sin A SIaNS AND VARIATIONS OF TRl@ONOMETRK FUNCTIONS 1 + + + + + + 0 to 1 1 to 0 0 to m CC to 0 1 to uz m to 1 - - II + + 1 to 0 0 to -1 -mtoo oto-m -cc to -1 1 to ca - III + + 0 to -1 -1 to 0 0 to d Cc to 0 -1to-m --CO to-1 - IV + - + - -1 to 0 0 to 1 -- too oto-m uz to 1 -1 to -- TRIGONOMETRIC FUNCTIONS 1 3 E X A C T V A L U E S F O R T R I G O N O M E T R I C F U N C T I O N S O F V A R I O U S A N G L E S Angle A Angle A in degrees in radians sin A COS A tan A cet A sec A csc A 00 0 0 1 0 w 1 cc 15O rIIl2 #-fi) &(&+fi) 2-fi 2+* fi-fi &+fi 300 ii/6 1 +ti *fi fi $fi 2 450 zl4 J-fi $fi 1 1 fi fi 60° VI3 Jti r 1 fi .+fi 2 ;G 750 5~112 i(fi+m @-fi) 2+& 2-& &+fi fi-fi 900 z.12 1 0 *CU 0 km 1 105O 7~112 *(fi+&) -&(&-Y% -(2+fi) -(2-&) -(&+fi) fi-fi 120° 2~13 *fi -* -fi -$fi -2 ++ 1350 3714 +fi -*fi -1 -1 -fi \h 150° 5~16 4 -+ti -*fi -fi -+fi 2 165O llrll2 $(fi- fi) -&(G+ fi) -(2-fi) -(2+fi) -(fi-fi) Vz+V-c? 180° ?r 0 -1 0 Tm -1 *ca 1950 13~112 -$(fi-fi) -*(&+fi) 2-fi 2 + ti -(&-fi) -(&+fi) 210° 7716 1 - 4 6 & l 3 f i - g f i -2 225O 5z-14 -Jfi -*fi 1 1 -fi -fi 240° 4%J3 -# -4 ti &fi -2 -36 255O 17~112 -&&+&Q -&(&-fi) 2+fi 2-6 -(&+?cz) -(fi-fi) 270° 3712 -1 0 km 0 Tm -1 285O 19?rll2 -&(&+fi) *(&-fi) -(2+6) -@-fi) &+fi -(fi-fi) 3000 5ïrl3 -*fi 2 -ti -*fi 2 -$fi 315O 7?rl4 -4fi *fi -1 -1 fi -fi 330° 117rl6 1 *fi -+ti -ti $fi -2 345O 237112 -i(fi- 6) &(&+ fi) -(2 - fi) -(2+6) fi-fi -(&+fi) 360° 2r 0 1 0 T-J 1 ?m For tables involving other angles see pages 206-211 and 212-215. f 14 TRIGONOMETRIC FUNCTIONS GF TRIGONOMETRIC FUNCTIONS | In each graph « is in radians. 5.22 y=snz 5.23 y = cos Fig. 5-5 Fig. 5-6 5.24 Yy = tang 5.25 y = ctz í Y | 4 Y ! ! Il | ' ' : | 1 k ; À ! ; I I | ; ! Í I 1 I 1 | 1 1 Í : I | I 1 | + Ea | í x E ÃO EE O EN IBN [5 3! z 12 ENTOSN! Pio Ly Pio: o I I J | |] Fig.5-7 Fig.58 5.26 y = secs 5.27 y = esex | Y 1 I y I 1 I ! | I I y | 1 ! | 1 1 I Í ! I ! No: | I 2 | I | | | | | I | I T ; I , | I ! I ! 2 | ! z -5 o E ” o (2m I I 1 | Í I I ! I I I 1 ! l t I 1 I, I i I I I I t ! i I I 1 i I I I I i I I I I 5.28 sin(-4) = —sinA 5.29 cos(-A) = cosA 5.30 tan(-4) = —tanA 5.31 cs(-4) = —cseA 5.32 sec(-A) = secÃ 5.33 cot(-4) = —eotA TRIGONOMETRIC FUNCTIONS 17 DIFRERÊNCE ANO PRODUCT FETRIC FUNCTIONS 541 snA +sinB = 2sinMA+B)cos!A-B) 5.62 sinÁ — snB — 2cos MA + B) sin KA — B) 5.63 cosA + cosB = 2cosHA+B)cosKA-—B) 5.64 csA — cosB = 2sinHA+B)smn MB-—4) 5.465 smAsinB = Heos(A-—B) — cos(A + B)) 5.66 cosA cosB = Meos(A—B) + cos(A+B)) 5.67 sinÁcosB = 4Msin(4 — B) + sin(A + B) 568 ainnd = ain4 (teosápi ("7 e cos Ajt=3 + ( Je cos Aju — e 569 cosa = À le cosAp — 22 cos Apt 4 (27 Bo eus aaa 2/ 146 ala cos , j njn—s — -F( > Je cosas 4 o 2n—1 1 x | ' 5.70 simíniA fsin(2n- DA — ( )simen-a +. E mA j 57 cost 1 A - 4 ) cos tn A + (mn eos u-mA +. a (ur )eosa) n-1/ 572º sinA = (20), DE Lssona — (EN cos En-DA + tom 24! . = ev) tamo (ue ns )eos / 573 cosmA = (1) + se [cosa + (er) cos (En BA + cr. + (11 Jeosza If «=siny then y=sin-lg, ie. the angle whose sino is à or inverse sine of x, is a many-valued funetion of « which is a collection of single-valued funetions called branches. Similavly the other inverse trigonometric functions are multiple-valued. For many purposes a particular branch is required, This is called the principal branch and the values for this branch are called principal valnes. 18 TRIGONOMETRIC FUNCTIONS oNoMerRIC:FU PRINCIPAL VAKUES É Principal values for & =: O Principal values for « < 0 0 =sincia = /2 7/2 E sin da <0 0 = costig E q/2 q/2 < cos ly E q 0=tanda< a/2 7/2 < tante < 0 O < cotria = : 7/2 < cotrla < q 0 = secla <s/2 m/2 < séc le E q O <csela E a/2 po 2 Eesele <0 j In all cases it is assumed that principal values are used. 5.74 sinviz icostix = q/2 5.80 sin-i(-2) = sing 5.75 tancla + cotrla = q/2 5.81 cosTl(=x) = q — costty 5.76 seca +ese te = q/2 5.82 tani(-5) = —tancle 577 eseciz = sini(1/0) 5.83 coti(-=a) = + —cot-lg 5.78 secla = cosTi(1/z) 5.84 seci(-a) = m— seclg 579 cotlz = tanc!(1/3) 5.85 cse(=a) = cesta In each graph y is in radians. Solid portions of curves correspond to principal values. 5.86 y = sincig 5.87 y = costig 5.88 v = tancisx / Y Y r a/2) = z z z +“ Jo à O f / =. = / —5/2 / - / f [= 19 5.89 y = cet-1% 5.90 y = sec-l% 5.91 y = csc-lx Fig. 5-14 Fig. 5-15 Fig. 5-16 TRIGONOMETRIC FUNCTIONS I Y _--/ T /’ /A-- / , --- -77 -- // , RElAilONSHfPS BETWEEN SIDES AND ANGtGS OY A PkAtM TRlAF4GlG ’ The following results hold for any plane triangle ABC with sides a, b, c and angles A, B, C. 5.92 Law of Sines a b c -=Y=- sin A sin B sin C 5.93 Law of Cosines A 1 /A C f with 5.94 Law with 5.95 cs = a2 + bz - Zab COS C similar relations involving the other sides and angles. of Tangents a+b tan $(A + B) - = tan i(A -B) a-b similar relations involving the other sides and angles. sinA = :ds(s - a)(s - b)(s - c) Fig. 5-1’7 where s = &a + b + c) is the semiperimeter of the triangle. Similar relations involving B and C cari be obtained. See also formulas 4.5, page 5; 4.15 and 4.16, page 6. angles Spherieal triangle ABC is on the surface of a sphere as shown in Fig. 5-18. Sides a, b, c [which are arcs of great circles] are measured by their angles subtended at tenter 0 of the sphere. A, B, C are the angles opposite sides a, b, c respectively. Then the following results hold. 5.96 Law of Sines sin a sin b sin c -z-x_ sin A sin B sin C 5.97 Law of Cosines cosa = cosbcosc + sinbsinccosA COSA = - COSB COSC + sinB sinccosa with similar results involving other sides and angles. 22 COMPLEX NUMBERS GRAPH OF A COMPLEX NtJtWtER A complex number a + bi cari be plotted as a point (a, b) on an xy plane called an Argand diagram or Gaussian plane. For example p,----. y in Fig. 6-1 P represents the complex number -3 + 4i. A eomplex number cari also be interpreted as a wector from 0 to P. * 0 - X Fig. 6-1 POLAR FORM OF A COMPt.EX NUMRER In Fig. 6-2 point P with coordinates (x, y) represents the complex number x + iy. Point P cari also be represented by polar coordinates (r, e). Since x = r COS 6, y = r sine we have 6.6 x + iy = ~(COS 0 + i sin 0) called the poZar form of the complex number. We often cal1 r = dm the mocklus and t the amplitude of x + iy. L - X Fig. 6-2 tWJLltFltCATt43N AND DtVlStON OF CWAPMX NUMBRRS 1bJ POLAR FtMM ilj 0” 6.7 [rl(cos el + i sin ei)] [re(cos ez + i sin es)] = rrrs[cos tel + e2) + i sin tel + e2)] 6.8 V-~(COS e1 + i sin el) ZZZ rs(cos ee + i sin ez) 2 [COS (el - e._J + i sin (el - .9&] DE f#OtVRtt’S THEORRM If p is any real number, De Moivre’s theorem states that 6.9 [r(cos e + i sin e)]p = rp(cos pe + i sin pe) . ” RCWTS OF CfMMWtX NUtMB#RS If p = l/n where n is any positive integer, 6.9 cari be written 6.10 [r(cos e + i sin e)]l’n = rl’n L e + 2k,, ~OS- + n where k is any integer. From this the n nth roots of a complex k=O,l,2 ,..., n-l. i sin e + 2kH ~ n 1 number cari be obtained by putting In the following p, q are real numbers, CL, t are positive numbers and WL,~ are positive integers. 7.1 cp*aq z aP+q 7.2 aP/aq E @-Q 7.3 (&y E rp4 7.4 u”=l, a#0 7.5 a-p = l/ap 7.6 (ab)p = &‘bp 7.7 & z aIIn 7.8 G = pin 7.9 Gb =%Iî/% In ap, p is called the exponent, a is the base and ao is called the pth power of a. The function y = ax is called an exponentd function. If a~ = N where a # 0 or 1, then p = loga N is called the loga&hm of N to the base a. The number N = ap is called t,he antdogatithm of p to the base a, written arkilogap. Example: Since 3s = 9 we have log3 9 = 2, antilog3 2 = 9. The fumAion v = loga x is called a logarithmic jwzction. 7.10 loga MN = loga M + loga N 7.11 log,z ; = logG M - loga N 7.12 loga Mp = p lO& M Common logarithms and antilogarithms [also called Z?rigg.sian] are those in which the base a = 10. The common logarit,hm of N is denoted by logl,, N or briefly log N. For tables of common logarithms and antilogarithms, see pages 202-205. For illuskations using these tables see pages 194-196. 23 24 EXPONENTIAL AND LOGARITHMIC FUNCTIONS NATURAL LOGARITHMS AND ANTILOGARITHMS Natural logarithms and antilogarithms [also called Napierian] are those in which the base a = e = 2.71828 18. . . [sec page 11. The natural logarithm of N is denoted by loge N or In N. For tables of natural logarithms see pages 224-225. For tables of natural antilogarithms [i.e. tables giving ex for values of z] see pages 226-227. For illustrations using these tables see pages 196 and 200. CHANGE OF BASE OF lO@ARlTHMS The relationship between logarithms of a number N to different bases a and b is given by 7.13 hb iv loga N = - hb a In particular, 7.14 loge N = ln N = 2.30258 50929 94.. . logio N 7.15 logIO N = log N = 0.43429 44819 03.. . h& N RElATlONSHlP BETWEEN EXPONBNTIAL ANO TRl@ONOMETRlC FUNCT#ONS ;; 7.16 eie = COS 0 + i sin 8, e-iO = COS 13 - i sin 6 These are called Euler’s dent&es. Here i is the imaginary unit [see page 211. 7.17 sine = eie - e-ie 2i 7.18 eie + e-ie case = 2 7.19 7.20 7.21 2 sec 0 = &O + e-ie 7.22 2i csc 6 = eie - e-if3 7.23 eiCO+2k~l = eie From this it is seen that @ has period 2G. k = integer HYPERBOLIC FUNCTIONS 27 AWMWM FORMWAS 0.2Q 8.21 8.22 8.23 sinh (x * y) = sinh x coshg * cash x sinh y cash (x 2 g) = cash z cash y * sinh x sinh y tanh(x*v) = tanhx f tanhg 12 tanhx tanhg coth (x * y) = coth z coth y 2 1 coth y * coth x 8.24 sinh 2x = 2 ainh x cash x 8.25 cash 2x = coshz x + sinht x = 2 cosh2 x - 1 = 1 + 2 sinh2 z 8.26 tanh2x = 2 tanh x 1 + tanh2 x HAkF ABJGLR FORMULAS 8.27 sinht = [+ if x > 0, - if x < O] 8.28 CoshE = cash x + 1 -~ 2 2 8.29 tanh; = k cash x - 1 cash x + 1 [+ if x > 0, - if x < 0] sinh x cash x - 1 Z ZZ cash x + 1 sinh x .4 ’ MUlTWlE A!Wlfi WRMULAS 8.30 sinh 3x = 3 sinh x + 4 sinh3 x 8.31 cosh3x = 4 cosh3 x - 3 cash x 8.32 tanh3x = 3 tanh x + tanh3 x 1 + 3 tanhzx 8.33 sinh 4x = 8 sinh3 x cash x + 4 sinh x cash x 8.34 cash 4x = 8 coshd x - 8 cosh2 x -t- 1 8.35 tanh4x = 4 tanh x + 4 tanh3 x 1 + 6 tanh2 x + tanh4 x 2 8 H Y P E R B O L I C F U N C T I O N S P O W E R S O F H Y P E R l 3 4 X A C & J f K l l O ~ S 8 . 3 6 s i n h z x = & c a s h 2 x - 4 8 . 3 7 c o s h z x = 4 c a s h 2 x + $ 8 . 3 8 s i n h s x = & s i n h 3 x - 2 s i n h x 8 . 3 9 c o s h s x = & c o s h 3 x + 2 c a s h x 8 . 4 0 s i n h 4 x = 8 - 4 c a s h 2 x + 4 c a s h 4 % 8 . 4 1 c o s h 4 x = # + + c a s h 2 x + & c a s h 4 x S U A & D I F F E R E N C E A N D F R O D U C T O F W P R R M 3 t A C F U k $ T l C W S 8 . 4 2 s i n h x + s i n h y = 2 s i n h & x + y ) c a s h $ ( x - y ) 8 . 4 3 s i n h x - s i n h y = 2 c a s h & x + y ) s i n h $ ( x - Y ) 8 . 4 4 c o s h x + c o s h y = 2 c a s h i ( x + y ) c a s h # - Y ) 8 . 4 5 c o s h x - c o s h y = 2 s i n h $ ( x + y ) s i n h $ ( x - Y ) 8 . 4 6 s i n h x s i n h y = * { c o s h ( x + y ) - c o s h ( x - y ) } 8 . 4 7 c a s h x c a s h y = + { c o s h ( x + y ) + c o s h ( x - ~ J } 8 . 4 8 s i n h x c a s h y = + { s i n h ( x + y ) - l - s i n h @ - Y ) } E X P R E S S I O N O F H Y P E R B O H C F U N C T I O N S ! N T E R M S O F ‘ O T H E R S I n t h e f o l l o w i n g w e a s s u m e x > 0 . I f x < 0 u s e t h e a p p r o p r i a t e s i g n a s i n d i c a t e d b y f o r m u l a s 8 . 1 4 t o 8 . 1 9 . s i n h x c a s h x t a n h x c o t h x s e c h x c s c h x s i n h x = u c o s h x = u t a n h x = u c o t h x = 1 1 t s e c h x = u c s c h x = w HYPERBOLIC FUNCTIONS 29 GRAPHS OF HYPERBOkfC FUNCltONS 8.49 y = sinh x 8.50 y = coshx 8.51 y = tanh x Fig. S-l Fig. 8-2 Fig. 8-3 8.52 y = coth x 8.53 y = sech x 8.54 y = csch x /i y 1 10 X 0 X -1 7 Fig. 8-4 Fig. 8-5 Fig. 8-6 Y \ L 0 X iNVERSE HYPERROLIC FUNCTIONS If x = sinh g, then y = sinh-1 x is called the inverse hyperbolic sine of x. Similarly we define the other inverse hyperbolic functions. The inverse hyperbolic functions are multiple-valued and. as in the case of inverse trigonometric functions [sec page 171 we restrict ourselves to principal values for which they ean be considered as single-valued. The following list shows the principal values [unless otherwise indicated] of the inverse hyperbolic functions expressed in terms of logarithmic functions which are taken as real valued. 8.55 sinh-1 x = ln (x + m ) -m<x<m 8.56 cash-lx = ln(x+&Z-ï) XZl [cash-r x > 0 is principal value] 8.57 tanh-ix = 8.58 coth-ix = X+l +ln - ( ) x-l x>l or xc-1 -l<x<l 8.59 sech-1 x O<xZl [sech-1 x > 0 is principal value] 8.60 csch-1 x = ln(i+$$G.) x+O 9 S O L U T I O N S o f A G E E M A I C E Q U A ’ I I O N S QUAURATIC EQUATION: uz2 + bx -t c = 0 9.1 S o l u t i o n s : -b 2 ~/@-=%c- x = 2a I f a , b, c a r e r e a l a n d i f D = b2 - 4 a c i s t h e discriminant, t h e n t h e r o o t s a r e ( i ) r e a l a n d u n e q u a l i f D > 0 ( i i ) r e a l a n d e q u a l i f D = 0 ( i i i ) c o m p l e x c o n j u g a t e i f D < 0 9.2 I f x r , x s a r e t h e r o o t s , t h e n x r + x e = -bla a n d x r x s = c l a . L e t 3a2 - a; Q = - - - - - - - 9 a r a s - 2 7 a s - 2 a f 9 ’ R= 5 4 , i Xl = S + T - + a 1 9.3 Solutions: x 2 = - & S + T ) - $ a 1 + + i f i ( S - T ) L x 3 = - - & S + T ) + a 1 - + / Z ( S - T ) I f a r , a 2 , a s a r e r e a l a n d i f D = Q3 + R2 i s t h e discriminant, t h e n ( i ) o n e r o o t i s r e a l a n d t w oc o m p l e x c o n j u g a t e i f D > 0 ( i i ) a 1 1 r o o t s a r e r e a l a n d a t l e a s t t w o a r e e q u a l i f D = 0 ( i i i ) a 1 1 r o o t s a r e r e a l a n d u n e q u a l i f D =C 0. I f D < 0, c o m p u t a t i o n i s s i m p l i f i e d b y u s e o f t r i g o n o m e t r y . 9.4 Solutions if D < 0: Xl = 2 a C O S ( @ ) x2 = 2 m C O S ( + T + 1 2 0 ’ ) w h e r e C O S e = -RI&@ x 3 = 2 G C O S ( + e + 2 4 0 ’ ) 9.5 xI + x2 + xs = - a r , x r x s + C r s x s + x s z r = Q , x r x 2 x s = - a s w h e r e x r , x 2 , x a a r e t h e t h r e e r o o t s . 32 SOLUTIONS OF ALGEBRAIC EQUATIONS 3 3 QUARTK EQUATION: x* -f- ucx3 + ctg9 + u 3 $ + a 4 = 0 Let y1 be a real root of the cubic equation 9.7 Solutions: The 4 roots of ~2 + +{a1 2 a; -4uz+4yl}z + $& * d-1 = 0 If a11 roots of 9.6 are real, computation is simplified by using that particular real root which produces a11 real coefficients in the quadratic equation 9.7. where xl, x2, x3, x4 are the four roots. - 10 FURMULAS fram Pt.ANE ANALYTIC GEOMETRY DISTANCE d BETWEEN TWO POINTS F’&Q,~~) AND &(Q,~~) 10.1 d= - Fig. 10-1 10.2 Y2 - Y1 mzz-z tan 6 F2 - Xl EQUATION OF tlNE JOlNlN@ TWO POINTS ~+%,y~) ANiI l%(cc2,1#2) 10.3 Y - Y1 Y2 - Y1 m cjr x - ccl xz - Xl Y - Y1 = mb - Sl) 10.4 y = mx+b where b = y1 - mxl = XZYl - XlYZ xz - 51 is the intercept on the y axis, i.e. the y intercept. EQUATION OF LINE IN ‘TEMAS OF x INTERCEPT a # 0 AN0 3 INTERCEPT b + 0 Y b a 2 Fig. 10-2 34 FORMULAS FROM PLANE ANALYTIC GEOMETRY 37 RQUATIQN OF’CIRCLE OF RADIUS R, CENTER AT &O,YO) 10.15 (a-~~)~ + (g-vo)2 = Re Fig. 10-10 RQUATION OF ClRClE OF RADIUS R PASSING THROUGH ORIGIN 10.16 T = 2R COS(~-a) Y where (r, 8) are polar coordinates of any point on the circle and (R, a) are polar coordinates of the center of the circle. Fig. 10-11 CONICS [ELLIPSE, PARABOLA OR HYPEREOLA] If a point P moves SO that its distance from a fixed point [called the foc24 divided by its distance from a fixed line [called the &rectrkc] is a constant e [called the eccen&&ty], then the curve described by P is called a con& [so-called because such curves cari be obtained by intersecting a plane and a cane at different angles]. If the focus is chosen at origin 0 the equation of a conic in polar coordinates (r, e) is, if OQ = p and LM = D, [sec Fig. 10-121 10.17 P CD T = 1-ecose = 1-ecose The conic is (i) an ellipse if e < 1 (ii) a parabola if e = 1 (iii) a hyperbola if c > 1. Fig. 10-12 38 FORMULAS FROM PLANE ANALYTIC GEOMETRY 10.18 Length of major axis A’A = 2u 10.19 Length of minor axis B’B = 2b 10.20 Distance from tenter C to focus F or F’ is C=d-- E__ 10.21 Eccentricity = c = - ~ a a 10.22 Equation in rectangular coordinates: (r - %J)Z + E = 3 a2 b2 0 Fig. 10-13 10.23 Equation in polar coordinates if C is at 0: re zz a2b2 a2 sine a + b2 COS~ 6 10.24 Equation in polar coordinates if C is on x axis and F’ is at 0: a(1 - c2) r = l-~cose 10.25 If P is any point on the ellipse, PF + PF’ = 2a If the major axis is parallel to the g axis, interchange x and y in the above or replace e by &r - 8 [or 9o” - e]. PARAR0kA WlTJ4 AX$S PARALLEL TU 1 AXIS If vertex is at A&,, y,,) and the distance from A to focus F is a > 0, the equation of the parabola is 10.26 (Y - Yc? = 10.27 (Y - Yo)2 = If focus is at the origin [Fig. 10.28 Fig. 10-14 Fig. 10-15 Fig. 10-16 4u(x - xo) if parabola opens to right [Fig. 10-141 -4a(x - xo) if parabola opens to left [Fig. 10-151 10-161 the equation in polar coordinates is 2a T = 1 - COS e Y Y -x 0 x In case the axis is parallel to the y axis, interchange x and y or replace t by 4~ - e [or 90” - e]. FORMULAS FROM PLANE ANALYTIC GEOMETRY 39 Fig. 10-17 10.29 Length of major axis A’A = 2u 10.30 Length of minor axis B’B = 2b 10.31 Distance from tenter C to focus F or F’ = c = dm 10.32 Eccentricity e = ; = - a (y - VlJ2 10.33 Equation in rectangular coordinates: (z - 2# os -7= 1 10.34 Slopes of asymptotes G’H and GH’ = * a 10.35 Equation in polar coordinates if C is at 0: a2b2 ” = b2 COS~ e - a2 sin2 0 10.36 Equation in polar coordinates if C is on X axis and F’ is at 0: r = Ia~~~~~O 10.37 If P is any point on the hyperbola, PF - PF! = 22a [depending on branch] If the major axis is parallel to the y axis, interchange 5 and y in the above or replace 6 by &r - 8 [or 90° - e]. 42 SPECIAL PLANE CURVES 11.18 Parametric equations: X = (a + b) COS e - b COS Y = (a + b) sine - b sin This is the curve described by a point P on a circle of radius b as it rolls on the outside of a circle of radius a. The cardioid [Fig. 11-41 is a special case of an epicycloid. Fig. 11-8 GENERA& HYPOCYCLOID 11.19 Parametric equations: z = (a - b) COS @ + b COS Il = (a- b) sin + - b sin This is the curve described by a point P on a circle of radius b as it rolls on the inside of a circle of radius a. If b = a/4, the curve is that of Fig. 11-3. Fig. 11-9 TROCHU#D 11.20 Parametric equations: x = a@ - 1 sin 4 v = a-bcos+ This is the curve described by a point P at distance b from the tenter of a circle of radius a as the circle rolls on the z axis. If 1 < a, the curve is as shown in Fig. 11-10 and is called a cz&ate c~cZOS. If b > a, the curve is as shown in Fig. ll-ll and is called a proZate c&oti. If 1 = a, the curve is the cycloid of Fig. 11-2. Fig. 11-10 Fig. ll-ll SPECIAL PLANE CURVES 43 TRACTRIX 11.21 Parametric equations: x = u(ln cet +$ - COS #) y = asin+ This is the curve described by endpoint P of a taut string PQ of length a as the other end Q is moved along the x axis. Fig. 11-12 WITCH OF AGNES1 11.22 Equation in rectangular coordinates: u = x = 2a cet e 11.23 Parametric equations: y = a(1 - cos2e) y = 2a Andy -q-+Jqx In Fig. 11-13 the variable line OA intersects and the circle of radius a with center (0,~) at A respectively. Any point P on the “witch” is located oy con- structing lines parallel to the x and y axes through B and A respectively and determining the point P of intersection. 8~x3 x2 + 4a2 l Fig. 11-13 11.24 11.25 11.26 11.27 il.28 FOLIUM OF DESCARTRS Equation in rectangular coordinates: x3 + y3 = 3axy Parametric equations: 1 3at x=m 3at2 y = l+@ Area of loop = $a2 \ 1 \ Equation of asymptote: x+y+u Z 0 Fig. 11-14 Y INVOLUTE OF A CIRCLE Parametric equations: I x = ~(COS + + @ sin $J) y = a(sin + - + cas +) This is the curve described by the endpoint P of a string as it unwinds from a circle of radius a while held taut. jY!/--+$$x . I Fig. Il-15 44 S P E C I A L P L A N E C U R V E S EVOWTE OF Aff ELLIPSE 11.29 E q u a t i o n i n r e c t a n g u l a r c o o r d i n a t e s : (axy’3 + (bvp3 = tu3 - by3 11.30 P a r a m e t r i c e q u a t i o n s : 1 c z z = ( C G - b s ) COS3 8 b y = ( a 2 - b 2 ) s i n s 6 T h i s c u r v e i s t h e e n v e l o p e o f t h e o r m a i s t o t h e e l l i p s e x e / a s + y z l b 2 = 1 s h o w n d a s h e d i n F i g .1 1 - 1 6 . F i g . 1 1 - 1 6 O V A L S OF CASSINI 1 1 . 3 1 P o l a r e q u a t i o n : f i + a4 - 2 a W ~ O S 2 e = b 4 T h i s i s t h e c u r v e d e s c r i b e d b y a p o i n t P s u c h t h a t t h e p r o d u c t o f i t s d i s t a n c e s r o m t w o f i x e d p i n t s [ d i s t a n c e 2 a a p a r t ] i s a c o n s t a n t b 2 . T h e c u r v e i s a s i n F i g . 1 1 - 1 7 o r F i g . 1 1 - 1 8 a c c o r d i n g a sb < a o r 1 > a r e s p e c t i v e l y . I f b = u , t h e c u r v e i s a Z e m k c a t e [ F i g . 1 1 - 1 1 . ++Y P _--- \ !--- a X F i g . 1 1 - 1 7 F i g . 1 1 - 1 8 LIMACON OF PASCAL 11.32 P o l a r e q u a t i o n : r = b + a c o s e L e t O Q b e a l i n e j o i n i n g o r i g i n 0 t o a n y p o i n t Q o n a c i r c l e o f d i a m e t e r a p a s s i n g t h r o u g h 0 . T h n t h e c u r v e i s t h e l o c u s o f a 1 1 p o i n t s P s u c h t h a t P Q = b . T h e c u r v e i s a s i n F i g . 1 1 - 1 9 o r F i g . 1 1 - 2 0 a c c o r d i n g a s b > a o r b < a r e s p e c t i v e l y . I f 1 = a , t h e c u r v e i s a c a r d i o i d [ F i g . 1 1 - 4 1 . - F i g . 1 1 - 1 9 F i g . 1 1 - 2 0 FORMULAS FROM SOLID ANALYTIC GEOMETRY 47 EQUATIONS OF LINE JOINING ~I(CXI,~I,ZI) AND ~&z,yz,zz) IN STANDARD FORM 12.5 x - x, Y - Y1 z - .z, x - Xl Y - Y1 ~~~~ or 2 - Zl % - Xl Y2 - Y1 752 - 21 1 =p=p m n These are also valid if Z, m, n are replaced by L, M, N respeetively. EQUATIONS OF LINE JOINING I’I(xI,~,,zI) AND I’&z,y~,zz) IN PARAMETRIC FORM 12.6 x = xI + lt, y = y1 + mt, 1 = .zl + nt These are also valid if 1, m, n are replaced by L, M, N respectively. ANGLE + BETWEEN TWO LINES WITH DIRECTION COSINES L,~I,YZI AND h , r n z , n z 12.7 COS $ = 1112 + mlm2 + nln2 GENERAL EQUATION OF A PLANE 12.8 .4x + By + Cz + D = 0 [A, B, C, D are constants] EQUATION OF PLANE PASSING THROUGH POINTS (XI, 31, ZI), (a,yz,zz), (zs,ys, 2s) x - X l Y - Y1 2 - .zl 12.9 xz - Xl Y2 - Y1 22 - 21 = cl x3 - Xl Y3 - Y1 23 - Zl or 12.10 Y2 - Y1 c! - 21 ~x _ glu + z2 - Zl % - Xl ~Y _ yl~ + xz - Xl Y2 - Y1 (z-q) = 0 Y3 - Y1 z3 - 21 23 - 21 x3 - Xl x3 - Xl Y3 - Y1 EQUATION OF PLANE IN INTERCEPT FORM 12.11 z+;+; z 1 where a, b,c are the intercepts on the x, y, z axes respectively. Fig. 12-2 48 FOkMULAS FROM SOLID ANALYTIC GEOMETRY E Q U A T I O N S O F L I N E T H R O U G H ( x o , y o , z c , ) A N D P E R P E N D I C U L A R T O P L A N E Ax + By + C.z + L = 0 x - X” Y - Yn P - 2 ” A z - z - B C or x = x,, + At, y = yo + Bf, z = .z(j + ct N o t e t h a t t h e d i r e c t i o n n u m b e r s f o r a l i n e p e r p e n d i c u l a r t o t h e p l a n e A x + B y + C z + D = 0 a r e A , B , C . D I S T A N C E F R O M P O I N T ( x e ~ , y , , , ~ ~ ) T O P L A N E AZ + By + Cz + L = 0 1 2 . 1 3 A q + B y , , + C z , , + D k d A + B z + G w h e r e t h e s i g n i s c h o s e n S O t h a t t h e d i s t a n c e i s n o n n e g a t i v e . N O R M A L F O R M F O R E Q U A T I O N O F P L A N E 1 1 2 . 1 4 x cas L + y COS,8 i- z COS y = p w h e r e p = p e r p e n d i c u l a r d i s t a n c e f r o m 0 t o p l a n e a t P a n d C X , / 3 , y a r e a n g l e s b e t w e e n O P a n d p o s i t i v e x , y , z a x e s . Fig. 12-3 T R A N S F O R M A T W N O F C O O R D l N A T E S I N V O L V I N G P U R E T R A N S L A T I O N 1 2 . 1 5 22 = x’ + x() x’ c x - x ( J y = y’ + yo o r y’ ZZZ Y - Y0 z = d + z ( J w h e r e ( % , y , ~ ) a r e o l d c o o r d i n a t e s [ i . e . c o r d i n a t e s r e l a - t i v e t o r y z s y s t e m ] , ( a ? , ’ , z ’ ) a r e n e w c o o r d i n a t e s [ r e l a - t i v e t o x ’ y ’ z ’ s y s t e m ] a n d ( q , 0 , z e ) a r e t h e c o o r d i n a t e s o f t h e n e w o r i g i n 0 ’ r e l a t i v e t o t h e o l d q z c o o r d i n a t e s y s t e m . ‘ X Fig. 12-4 FORMULAS FROM SOLID ANALYTIC GEOMETRY 49 TRANSFORMATION OF COORDINATES INVOLVING PURE ROTATION x = 1 1 x 1 + & y ! + 1 3 % ’ \ % ’ 12.16 y = WQX’ + wtzyf + r n p ? \ \ 2 = n l x ' + n 2 y ' + n 3 z ' \ \ X ' = Z I X + m 1 y + T z l Z \ \ \ O l ? i y' = 1 2 x + m 2 y + n p . x ' = z z x + m a y + ? % g z where the origins of the Xyz and x’y’z’ systems are the * , ?/‘ , , , Y ’ , 1 ’ 3 ’ ~ Y same and li, ' m l , n l ; 1 2 , m 2 , n 2 ; 1 3 , m 3 , n s are the direction cosines of the x’, ,y’, z’ axes relative to the x, y, .z axes ,,/ X respectively. Fig. 12-5 TRANSFORMATION OF COORDINATES INVOLVING TRANSLATION AND ROTATION z = Z I X ’ + & y ’ + l& + x. z 12.17 F’ y = miX’ + mzy’ + ma%’ + yo ' \ \ , y 1 = n l X ' + n 2 y ' + n 3 z ' + z . l , 2 \ , / ' i X ' = 4 t x - X d + m I t y - y d + n l b - z d o r $ " , ? / , ) > q J l or y! zz &z(X - Xo) + mz(y - yo) + n& - 4 / x’ = &(X - X0) + ms(y - Y& + 42 - zO) / - Y / where the origin 0’ of the x’y’z’ system has coordinates (xo, y,,, zo) relative to the Xyz system and Zi,mi,rri; la, mz, ‘nz; &,ms, ne are the direction cosines of the X’, y’, z’ axes relative to the x, y, 4 axes respectively. ‘X’ Fig. 12-6 CYLINDRICAL COORDINATES (r, 0,~) A point P cari be located by cylindrical coordinates (r, 6, z.) [sec Fig. 12-71 as well as rectangular coordinates (x, y, z). The transformation between these coordinates is x = r COS0 12.18 y = r sin t or 0 = tan-i (y/X) z=z Fig. 12-7 5 2 FORMULAS FROM SOLID ANALYTIC GEOMETRY H Y I ’ E R B O L O I D O F T W O S H E E T S Note orientation of axes in Fig. 12-14. Fig. 12-14 E L L I P T I C P A R A B O L O I D 1 2 . 3 0 Fig. 12-15 H Y P E R B O l f C P A R A B O L O I D 1 2 . 3 1 xz y2 z --- = _ a2 b2 C Note orientation of axes in Fig. 12-16. / X - Fig. 12-16 D E F t N l l l O N O F A D E R t V A T l V R If y = f(z), the derivative of y or f(x) with respect to z is defined as 13.1 ~ = lim f(X+ ‘) - f(X) = d X h a i r f ( ~ + A ~ ) - f ( ~ ) h + O Ax-.O Ax where h = AZ. The derivative is also denoted by y’, dfldx or f(x). The process of called di#e~eAiatiotz. taking a derivative is G E N E R A t . R l t k E S O F D t F F E R E t W t A T t C W In the following, U, v, w are functions of x; a, b, c, n are constants [restricted if indicated]; e = 2.71828. . . is the natural base of logarithms; In IL is the natural logarithm of u [i.e. the logarithm to the base e] where it is assumed that u > 0 and a11 angles are in radians. 1 3 . 2 1 3 . 3 1 3 . 4 1 3 . 5 1 3 . 6 1 3 . 7 1 3 . 8 1 3 . 9 1 3 . 1 0 1 3 . 1 1 1 3 . 1 2 1 3 . 1 3 g(e) = 0 &x) = c & c u ) = c g & u v ) = u g + v g $-(uvw) = 2 dv du uv- + uw- + vw- dx dx du _ -H - v(duldx) - u(dv/dx) dx v V Z - & n j z z & $ du _ dv du - - ijii - du dx (Chai? rule) du 1 -=- dx dxfdu dy dyidu z = dxfdu 5 3 54 DERIVATIVES AL”>. 1 _. .i ” ., d 13.14 -sinu = du dx cos YG du 13.17 &cotu = -csck& 13.15 $cosu = -sinu$ 13.18 &swu = secu tanus 13.16 &tanu = sec2u$ 13.19 -&cscu = -cscucotug 13.20 -& sin-1 u =$=$ -%< sin-‘u < i 1 13.21 &OS-~, = -1du qciz dx [O < cos-lu < z-1 13.22 &tan-lu = LJ!!+ 1 + u2 dx C -I < tan-lu < t 1 13.23 &cot-‘u = +& [O < cot-1 u < Tr] 13.24 &sec-‘u = 1 du if 0 < set-lu < 7712 ju/&zi zi = I if 7712 < see-lu < r 13.25 & - csc-124 = if 0 < csc-l u < 42 + if --r/2 < csc-1 u < 0 1 d l’Xae du 13.26 -log,u = ~ - dx u dx a#O,l 13.27 &lnu = -&log,u = ig 13.28 $a~ = aulna;< 13.29 feu = d" TG fPlnu-&[v lnu] = du dv vuv-l~ + uv lnu- dx 13.31 gsinhu = eoshu:: 13.32 &oshu = du sinh u dx 13.34 2 cothu = - cschzu ;j 13.35 f sech u = - sech u tanh u 5 dx 13.33 $ tanh u = sech2 u 2 13.36 A!- cschu = dx - csch u coth u 5 dx If 2 = f(z), then y is the function whose derivative is f(z) and is called the anti-derivative of f(s) or the indefinite integral of f(z), denoted by s f (4 dx. Similarly if y = S f (4 du, then $ = f(u). Since the derivative of a constant is zero, all indefinite integrals differ by an arbitrary constant. For the definition of a definite integral, see page 94. The process of finding an integral is called integration. In the following, u, v, w are functions of x; a, b, p, q, n any constants, restricted if indicated; e = 2.71828. . . is the natural base of logarithms; In u denotes the natural logarithm of u where it is assumed that u > 0 [in general, to extend formulas to cases where u < 0 as well, replace In u by In ]u]]; all angles are in radians; all constants of integration are omitted but implied. 14.1 S adz = ax 14.2 14.3 14.4 S uf(x) dx = a S f(x) dx S (ukz)kwk . ..)dx = _(‘udx ” svdx * .(‘wdx * ... S udv = WV - S vdu [Integration by parts] For generalized integration by parts, see 14.48. 14.5 S 1 f(m) dx = - a S f(u) du 14.6 S F{fWl dx = S F(u)2 du = S F(u) f’(z) du where u = f(z) 14.7 S .&a+1 undu = - n-t 1’ n#-1 [For n = -1, see 14.81 14.8 S du -= In u U if u > 0 or In (-u) if u < 0 = In ]u] 14.9 14.10 S eu du = eu s audu = S @Ina& = eUl”Ll au -=- In a In a ’ a>O, a#1 57 58 INDEFINITE INTEGRALS 14.11 I‘ sinu du = - cos u cosu du = sin u 14.13 I‘ tanu du = In secu = -In cosu 14.14 cot u du = In sinu 14.15 see u du = In (set u + tan u) = In tan 14.16 I‘ csc u du = ln(cscu- cotu) = In tan; 14.17 .I' sec2 u du = tanu 14.18 * I csc2udu = -cotu 14.19 S tanzudu = tanu - u 14.20 S cot2udu = -cotu - u 14.21 S U sin 2u sin2udu = - - - = 2 4 #u - sin u cos u) 14.22 ' s co532 u du = sin 2u ;+T = j&u + sin u cos u) 14.23 S secutanu du = secu 14.24 s cscucotudu = -cscu 14.25 S sinhu du = coshu 14.26 I‘ coshu du = sinh u 14.27 I‘ tanhu du = In coshu 14.28 J coth u du = In sinh u 14.29 S sechu du = sin-1 (tanh u) or 2 tan-l eU 14.30 S csch u du = In tanh; or - coth-1 eU 14.31 J sechzudu = tanhu 14.32 I‘ csch2 u du = - coth u 14.33 s tanh2u du = u - tanhu INDEFINITE INTEGRALS 14.34 S cothe u du = u - cothu 59 14.35 S sinh 2u u sinheudu = --- = 4 2 +(sinh u cash u - U) 14.36 S sinh 2u coshs u du = ___ 4 i- t = Q(sinh u cash u + U) 14.37 S sech u tanh u du = - sech u 14.38 s csch u coth u du = - csch u 14.39 ___ = S du u’ + CL2 14.40 S u2 > a2 14.41 S - = u2 < a2 14.42 s 14.43 ___ s du @T7 = ln(u+&Zi?) 01‘ sinh-1 t 14.44 14.45 14.46 14.47 14.48 S f(n)g dx = f(n-l,g - f(n-2)gJ + f(n--3)gfr - . . . (-1)” s fgcn) dx This is called generalized integration by parts. Often in practice an integral can be simplified by using an appropriate transformation or substitution and formula 14.6, page 57. The following list gives some transformations and their effects. 14.49 S 1 F(ax+ b)dx = - a S F(u) du where u = ax + b 14.50 S F(ds)dx = i S u F(u) du 14.51 S F(qs) dx = f S u-1 F(u) du where u = da where u = qs 14.52 S F(d=)dx = a S F(a cos u) cos u du where x = a sin u 14.53 S F(dm)dx = a S F(a set u) sec2 u du where x = atanu 62 14.89 14.90 14.91 14.92 14.93 14.94 14.95 14.96 14.97 14.98 14.99 14.100 14.101 14.102 14.103 14.104 INDEFINITE INTEGRALS s dzbdx = “7 s xd-6 dx = 2(3a;z; 2b’ l&a@ s x%/G dx = 2(15a’x2 ;$a;bx + 8b2) ,,m3 J ‘&zT dx = 2d&3 + b s dx x&zz [See 14.871 s &dx = &zTT dx x2 X +; s X&iZT [See 14.871 s &T” = 2LlFqz 2mb (2m + 1)~ - (2m + 1)a s dXGb dx s dx \/azfb = - _ (2m - 3)a X+GT3 (m - l)bxm-1 (2m - 2)b s x-l:= s xmd= dx = t2;$8,, (as + b)3’2 - c2;“+b3,a s X~-QL-TTdX .(‘ &iTx &&x5 Xm dx = - (m-l)xm-’ + 2(mf 1) s x--l:LTT s l/zT-ii -----dx = -(ax + b)3/2 _ (2m - 5)a s &GTT Xm (m - l)bxm-’ (2m - 2)b gm--1 dx c (ax + b)m’2 dx = 2(ax + b)(“‘+z)lz a(m + 2) s x(ax + b)““z dcv s z2(ax + b)m’2 dx = 2(ax + b)(“‘+Q/z 2b(ux + b)(m+z)/z a2(mf4) - aym + 2) = 2(ax + b)(m+s)lz _ 4b(ux + b)(m+4)/2 u3(m -I- 6) a3(m+ 4) + 2b2(ax + b)(“‘+2)‘2 u3(m+ 2) s (as +xbP”2 dx = ~(CLX + b)““z + b s (ax + b)(m-2)/2 dx m X S (ax + b)m’z dx = - (ax + b)(m+2)‘2 X2 bx +z S (ax + b)m’2 dx X S dx 2 1 S dx x(ux + b)m/2 = (m - Z)b(ax + b)(m-2)/2 ’ 5 x(ax + b)(“‘--2)/z INTEGRALS INVOLVfNC c&z + b AND p;z! + q >: “: 14.105 dx (ax + b)(w + d 14.106 (‘ x dx . (ax + b)(px + d = & g In (ax+ b) - % In (px+ q) 14.107 S dx (ax + b)2bx + d 14.108 S xdx (ax + b)2(px + 4 x2 ds 14’109 j- (ax + b)z(px + q) = (bp - aq;&ux+ b) + ’ (b- ad2 b(bp ,Z 2uq) In (uz + b) 14.110 14.111 14.112 14.113 14.114 63 INDEFINITE INTEGRALS I’ dx -1 1 (ax + bpqpx + qp = (Yz - l)(bp - aq) 1 (ax + b)+l(pz + q)“-’ + a(m+n-2) s dx s ax + b = 7 + yh(px+q) (ax + bpqpx + q)n-1 -dds PX + Q I -1 (N - l)(bp - uq) 1 (ax + bp+’ (px + q)“-l + (x-m - va s ,E++q;!Tl dx > s (ax + bp (px+ q)n dx = -1 { (ax + bp (m - m - l)p (px + q)n-l + m@p - aq) s (ax + b)m- 1 (px+ 4” dx > (n--:)p i (ax + ap (pxtqy-1 - mu S (ax + by-1 \ (px + qy- l dx1 S -E!C&.Y dx = 2(apx+3aq-2bp)Gb d&zT 3u2 s dx (Px + 9) &ii-G 14.115 Jgdx = 14.116 s (px + q)” dn~ dx 2(px + q)n+ l d&T? = (2n + 3)P b - aq I (2n + 3)P s (Px + q)” dx dn 14.117 S dx daxi-b = dx (px + 9)” &z-i (n - l)(aq - bp)(px + q)n-l + 2(n ‘“^I),;)” bp) s (px + q)n-1 &-TT 14.118 - S bx + dn dx = 2(px + q)n &iTT * (px + q)“- l dx da (2n + 1)u + 2n(aq - W (2n + 1)a s &ii% &zTiT 14.119 Smdx = -&m (n - l)p(pz + qy- l + 2(n ” 1)p s 1 dx (px + qp-’ ~GzT INTEBRAES INVOLVING ds AND J/K 14.120 S, dx ZI i &ln(dGFG+~) (ax + b)(w + q') 14.121 xdx dbx + b)(px + 4 = b + w dx (ax + b)(px + q) UP --x&T- (ax + b)(w + q) 64 INDEFINITE INTEGRALS 14.122 (ax + b)(px + q) dx = . 14.123 .(' j/sdx = ‘@‘+ y(px+q) + vj- (ax+;(px+q) 2&izi 14.124 (aq - W d%=i dx (ax + b)(px + 4 lNTEGRALS INVOLVtNO x’ + a2 14.125 s--$$ = $I-'~ 14.126 J-$$$ = + In (x2 + a2) 14.127 J$$ = x - a tan-13c a 14.128 s& = $ - $ln(x2+az) ‘4-l 30 J x2(x?+ ($2) = --- six +3 tan-l: 14.131 J x3(x?+a2) = 1 -- 2a2x2 14.132 J (x2d;Ga2)2 = 2a2(xf+ a2) + &3 tan-': 14.137 S dx 1 x x2(x2 + c&2)2 = --- -- a4x 2a4(x2 + a2) 2:5 tan-l: 14.138 S dx 1 1 x3(x2 + a2)2 = -~ - 2a4x2 2a4(x2 + u2) S (x2d+za2)n = X 2n - 3 S dx 14.139 2(n - l)a2(x2 + a2)%-* + (2n- 2)a2 (x2 + a2)n-1 14.140 S xdx -1 (~2 + a2)n = 2(n - 1)(x2 + a2)n-1 14.141 S dx 1 x(x2 + a2)” = 2(12 - l)a2(x2 + uy--1 + $ S dx x(x2 + a2)n-1 14.142 xm dx xm--2 dx x*--2 dx . (x2+ a2)" = S (x2 + a2)n-l - a2 S (x2 + a2)" 14.143 S dx 1 dx 1 dx x9z2+a2)n = 2 S -- 33x2 + a2)n--1 a2 S xme2(x2 + a2)” INDEFINITE INTEGRALS 14.182 14.183 14.184 14.185 14.186 14.187 14.188 14.189 14.190 14.191 14.192 14.193 14.194 14.195 14.196 14.197 14.198 14.199 14.200 14.201 14.202 S x dx ___ II ~~ S x2 dx - = lfzT-2 S x3 dx ~I2xz = S In (x + &&?) or sinh-1s a x 7 a2 2 +a -- 2 2 In(x+@Tz) (x2 + a2)3/2 3 - a2&GZ S J/X- .2&F&i = - a22 s dx = ~~ + k3 In a+&3T2 x3~~5 -2a2x2 X > S + $l(x+~W) S xdmdx = (x2 + a2)3/2 3 s x%jmdx = x(x2 + a2)312 a2x&T2 a4 4 - 8 sln(x+j/~) S ad-g-q dx = (x2 + a2)5/2 _ a2(x2 + a2)3/2 5 3 s = &qgwalIn S &T &G-G -dx = -- X2 + ln(z+drn) S &s-T-z - $a In a-l-&372 x3 X S dx (x2 + a2)3/2 = s x dx (%2 + a2)3/2 = &is f x2 dx . (x2 + a2)3/2 = d& + ln(x + d&i7) s x3 dx (x2 + a2)3/2 = im+a2 @TTP s dx x(x2 + a2)3/2 = 1 a2&SiZ - f In ( a+JZ2 2 S dx ~~ x x2(x2 + a2)3/2 = - ~ - a4x a4&FS S dx -1 3 3 a+&-TS x3(x2 + a2)3/2 = - 2a2x2&GT 2a4&FiZ + s5ln 2 68 INDEFINITE INTEGRALS 14.203 S (x2 + a~)312 dx = x(x2 + u2)3/2 3&q/~ 4 + 8 +~a4ln(x+~2TTq 14.204 S x(x2 + u2)3/2 dx = (x2 + u2)5/2 5 14.205 S x2(x2 + ~2)3/2 ds = x(x2 + u2)5/2 _ u2x(x2 + u2)3/2 u4x@TF2 24 - -- 6 16 ~~ln(~+~2xq 14.206 S x3(x2 + u2)3/2 dx = (22 + ~247’2 ~2(~2 + ~2)5/2 7 - 5 14.207 S (x2 + u2)3/2 dx = (x2+ u2)3’2 CL+@-TT? X 3 + u2@T2 - a3 In x > 14.208 S (x2 + UT’2 ds = x2 _ (x2 + u2)3’2 + 3x- + 3a2 ln (x + q-&-T&) x 2 2 14.209 S (x2 + U2)3’2 (x2 + a2)3/2 dx = - 2x2 U-kdlXS x3 x > 14.210 14.211 14.212 14.213 14.214 14.215 14.216 14.217 14.218 s In (x + j/277), S S x2 dx 5 x-a P-- ~ = &G=z 2 ’ x3dx s G= 1 5 x2- u2 asec-l X I I U S x3(& = @=2 2u2x2 + k3 see-l x I I U s dndx = 7 x x2-a -$ln(x+dm) S xda~dx = (x2 _ u2)3/2 3 S x2@73 dx = x(x2 - a2)3/2 cAq/m~ -- 4 + 8 “8” ln(x + +2TS) 14.219 S ,“d~ dx = cx2 - ~2)5/2 + ~2(~2 - ~2)3/2 5 3 14.220 s-dx = dm- a see-l - I;1 14.224 14.225 14.226 14.227 14.228 14.229 14.230 14.231 14.232 14.233 14.234 14.235 14.236 INDEFINITE INTEGRALS S 22 dx (~2 - a2)3/2 = -~ &z + ln(x+&272) S x3 dx (22 - a2)3/2 = GTZ- - dx2aLa2 S dx -1 4x2 - a2P2 = a2@qp 1 -- a3 set-1 2 I I a S dx lJZ2 x z2(s2 - a2)3/2 = -_ - a4x a+iGZ S dx 1 = 3 3 x3(x2 - a2)3/2 -- 2a5 see-l : I I a S (~2 - &)3/z & z x(x2 - a2)3/2 3a2x&iF2 4 - 8 + I * x(52 - a2)3/2 dx = (x2 - a!2)5/2 5 S x2(99 - a2)3/2 dx = 2(x2 - a2)5/2 a2x(x2 - a2)3/2 6 + 24 S x3(52 - a2)3/2 dx = (22 - a2)7/2 7 + az(x2 - a2)5/2 5 :a4 In (5 + &372) a4x&FS - 16 + $ In (z + $X2 - a2 ) S @2 _ a2)3/2 X dx = tx2 - a2)3'2 - a2da + a3 set-' c 3 I I S (x2 _ a2)3/2 X2 dx = - (x2 -xa2)3'2 + 3xy _ ia ln (1 + da) S @2 - a2)3/2 x3 ,jx = _ (x2;$33'2 + "y _ ga sec-' [El a 69 1NtEORAtS lNVC)LVlNG <%=?? 14.237 14.238 14.239 14.240 14.241 14.242 14.243 S da& = sin-l: xdx ____ = -dGi @G? S x2 dx x a-x 7 ___ = - ).lm 2 s x3 dx ____ = jlzz (a2 - x2j312 _ a2dpz3i 3 a+&KG X S dx x743x5 a + I/-X; -~ 2a2x2 - &3 In 5 72 INDEFINITE INTEGRALS In the following results if b2 = 4ac, \/ ax2 + bx + c = fi(z + b/2a) and the results on uaaes 60-61 can be used. lf b = 0 use the results on pages 67-70. If a = 0 or c = d use the results on pages 61-62. 14.280 ax $ In (2&dax2 + bx + e + 2ax + b) a = ax2+bx+c -&sin-l (J;rT4ic) or & sinh-l(~~~c~~2) 14.281 14.282 s, x2 dx ax2+bx+c 14.283 14.284 dx = - ax2 + bx + c 14.285 14.286 ax2+bx+cdx = (2ax+ b) ax2+ bx+c 4a +4ac-b2 dx 8a . ax2 + bx + c = (ax2 + bx + c)3/2 3a ~ ax2+ bx+c b(2ax + b) dp - 8a2 b(4ac - b2) dx - 16a2 axz+bx+c 14.287 = 6az4a25b (ax2 + bx + c)~/~ + “““,,,“” J d ax2f bx+c dx 14.288 S“ ax2+bx+c X 14.289 ax2+bx+c X2 14.290 S dx (ax2 + bx + c)~‘~ 14.291 S x dx (ax2 + bx + ~)3’~ 14.292 S x2 dx cax2 + bx + 43’2 2(2ax + b) (4ac - b2) ax2 + bx + c 2(bx + 2c) (b2 - 4ac) \/ ax2 + bx + c (2b2 - 4ac)x + 2bc a(4ac - b2) 1~x2 + bx + c dx ax2+bx+c 14.293 14.294 14.295 S +x2 +% + c)3’2 = cdax2 : bx + e + : SJ dx x axz+bx+c S (QX~ + ifif + 4312 s dx ax2 + 2bx + c - &?xdax2 + bx + c + b2 - 2ac S dx x2(aX2 + bx + c)~‘~ = 26 cax2 + bx + 43’3 3b S, dx -- 2c2 x ax2+bx+c S (ax2 + bx + c)n+1/2dx = (2ax + b)(ax2 + bx + c)n+ 1~2 4a(nf 1) + (2% + 1)(4ac- b2) 8a(n+ 1) S (a&+ bx + c)n-1’2dx . INDEFINITE INTEGRALS 73 14.296 S x(uxz + bx + C)n+l/z dx = (ax2 + bx + C)n+3'2 _ $ cq2n+ 3) s (ax2 + bx + ~)~+l’zdx . 14.297 ’ s dX (ax2-t bx + ~)n+l’~ = 2(2ax + b) (2~2 - 1)(41x - b2)(ax2 + bx + +--1/z 8a(n- 1) + (2~2 - 1)(4ac - b2). (‘ dx (61.x2 + bx + c)n--1E 14.298 s dx 1 x(ux2 + bx + ++I’2 = (2~2 - l)c(ux’J + bx + c)n--1’2 s dx x(ux2 + bx + c)“-~‘~ 2”~ s dx -- (ax2 + bx + c),+ l/i JPJTEORALS JNVOLVING 3ea + a3 Note that for formulas involving x3 - u3 replace a by --a. 14.299 ~ = s dx X3 + u3 14.300 ~ = s x dx x3 + a3 x2 - ax + cl2 x2 - ax + c-9 2x-u (x + c-42 + 1 tan-l 7 a\/3 43 14.301 __ = s x2 dx x3 + CL3 $ In (x3 + ~3) 14.303 s ClX 1 1 x2(x3 + u3) = -- - a32 G-4 14.304 .( '(z3yu3)2 = X 3u3(s3 +a3) + 14.305 s ' xdx (x3 + c&3)2 x2 + = 3a3(x3 + a3) 14.306 s x2 dx (x3+ u3)2 = 1 - 3(x3 + US) 14.302 s dx x(x3+u3) = In x2 - ax + u2 (x + a)2 - +3tanP1 F &In (xfcp + 2 2x-u x2 - ax + a2 - tan-l - 3u5fi a 3 \r &n x2 - ax + a2 2x - a (x + a)2 + 3utfi3 tan-’ 3 14.307 s dx 1 %(X3 + a3)2 = &,3(x3 + as) 14.308 s dx 1 x dx x2(x3 + u3)2 = -- - x2 -4-.--- CL62 3a6(x3 + u3) 3u6 s x3 + u3 [See 14.3001 14.309 s x-’ dx xm-2 ~ = - - a3 xm-3 dx ~ x3 + u3 m-2 x3 + a3 14.310 s dX -1 -2 s dx x9x3+ a3) = c&3@- 1)x+-’ u3 xn-3(x3 + u3) JNTEORALS INVOLYJNG c?+ * a* 14.311 - = I' dx 1 1 - In x2 + axfi + a2 -- tan-1 -!!tC-L T x4 + a4 4u3fi x2 - uxfi + c&2 2aqi 22 - CL2 14.312 ~ = S xdx x4 + u4 & tan-l $ -L In x2 - axfi + a2 1 -- tan-1 -!!G!- 6 4ufi x2 + ax& + u2 2ckJr2 x2 - a2 14.314 ~ = S x3 dx x4 + a4 $ In (x4 + a4) 74 INDEFINITE INTEGRALS 14.315 s dx x(x4 + d) 14.316 s dx 1 x2(x4 + u4) = tan-l CiXfi +- ___ 2a5& x2 - a2 14.317 dx x3(x4 + a4) = . 14.322 14.323 14.324 14.325 .I’ dx x(xn+an) = &nlnz xn + an 14.326 fs = ‘, In (29 + an) 14.327 S xm dx s xm--n dx (x”+ c&y = (xn + (yy-l - an s x”’ --n dx (xn + an)T 14.328 I’ dx 1 dx 1 dx xm(xn+ an)’ = 2 s xm(xn + IP)~--~ -s an xmpn(xn + an)r 14.332 x”’ dx s-- = an (xn - an)’ S xm--n dx (~“-a~)~ + s xm--n dx (xn-an)r-l 14.333 = 14.334 S dx = m..?wcos-~ !qfzGG m/z . INDEFINITE INTEGRALS 77 14.369 ' cosax dx = * a 14.370 s cos ax x sin ax xcosaxdx = - ~ a2 + a 14.371 - xzcosaxdx = $,,,a. + sin ax 14.372 ' x3 cosax dx = (T---$)cosax + ($-$)sinax 14.373 s (axY kd4 Fdx = Ins-- -- 2*2! + 4*4! (axF -+ . . . 6*6! 14.374 ";,' dx = - cos ax _ a X S' 'y dx [See 14.3433 14.375 --GE-= = cos ax $ In (see ax + tan ax) 14.376 - = S x dx En(ax)2n + 2 cos ax (2n-k2)(2n)! + ... 14.377 s co532 ax dx = sin 2ax f+- 4a 14.378 x co9 ax dx = x sin 2ax cos 2ax -+- 4a 8cG 14.379 s cos3 ax dx = sin ax sin3 ax - - a 3a 14.380 cos4 ax dx = dx tan ax 14.381 ___ = - s COG ax a 14.382 dx - = cos3 ax 14.383 cos ax cos px dx = sin (a - p)x 2(a - P) + sin (a + p)x 2(a + P) [If a = *p, see 14.377.1 14.384 s dx = 1 - cosax 14.385 s x dx x 2 -- cot E + - In sin ax 1 - cos ax = a 2 a2 2 14.386 dx = 1 + cosax 14.387 xdx = 1 + cos ax 14.388 JtI _ dx cos ax)2 = dx 14’389 S (1 + cosax)2 = 78 INDEFINITE INTEGRALS 14.390 s dx p+qcosax = I ad-2tan-’ dt/(p - Mp + 4 tan ?px [If p = *q see &j&2 In ! tan *ax + d(q + dl(q -PI 14.384 and 14.386.1 tan &ax - d(q + dl(q - P) 14.391 s dx (p + q cos ax)2 = q sin ax P -- a($ - $)($I + q cos ax) 42 - P2 s dx [If p = *q see p + q cos ux 14.388 and 14389.1 14.392 s dx 1 p2 + q2 cos2 ax = w/FS tan-l P tan ax dn7 14.393 s dx p2 - q2 cos2 ax = I ap + tan-l E p2- q2 1 WdFT2 In ptanax-dm ( ptanax+dv > 14.394 s xm cos ax dx = xm sin ax mxm--l +- a a2 cos ax - mtm - 1) a2 S xm-2 cos ax dx 14.395 s ydx = - cos ax a -- (n - 1)x*- 1 n-1 S’ sdx [See 14.3651 14.396 s co@ ax dx = sin ax cosn--I ax +?Z-1 an - s co@-2ax dx n 14.397 s .-AL= sin ax dx co@ ax a(n - 1) co@--I ax +n-2 -s n-l COP-2 ax S xdx 14.398 - = x sin ux 1 xdx - COP ax a(n - 1) COP--I ax a2(n - l)(n - 2) cosnP2 ax +n-2 -s n-1 cosn-2 ax 14.399 S sin2 ax sinax cosax dx = - 2a 14.400 S sin px cos qx dx = _ cos (P - q)x _ cos (P + q)x VP - 4 VP + 9) 14.401 s sinn ax cos ax dx = sinn + 1 ax (n + 1)~ [If n = -1, see 14.440.1 14.402 S COP ax sin ax dx = -cosnflax (n + 1)a [If n = -1, see 14.429.1 14.403 S X sin 4ax sin2 ax cos2 ax dx = - - - 8 32a 14.404 S dx =1 sin ax cos ax a In tan ax 14.405 S dx = A In tan 1 sin2 ax cos ax a a sin ax 14.406 S dx =1 sin ax ~052 ux ;lntan y + & 14.407 S dx = -2cot2ax sin2 ax cos2 ax a 14.411 dx - k 1 . sinax(1 2 cosax) - 2a(l * cos ax) 14.412 S dx sin ax rfr cos ax L In tan = a& 14.413 sin ax dx = sin ax * cos ax I T $a In (sin ax * cos ax) 14.414 s cos ax dx = sin ax f cos ax 2: + +a In (sin ax C cos ax) 14.415 sin ax dx p+qcosax = - $ In (p + q cos ax) 14.416 cos ax dx p+qsinax = $ In (p + q sin ax) 14.417 S sin ax dx = 1 (p + q cos axy aq(n - l)(p + q cos axy-1 18 s cos ax dx -1 (p + q sin UX)~ = aq(n - l)(p + q sin UX)~--~ 14.4 14.4 19 dx = adi+ q2 In tan ax + tan-l (q/p) p sin ax + q cos ax 2 2 a&2-p2-q2tan-1 p + (r - q) tan (ax/z) 14.420 dx T2 - p2 - q2 p sin ax + q cos ax + T = 1 ln p - dp2 + q2 - r2 + (r - q) tan (ax/2) - aVp2 + q2 - ~-2 p + dp2 + q2 - r2 + (T - q) tan (ax/2) If r = q see 14.421. If ~~ = p2 i- q2 see 14.422. INDEFINITE INTEGRALS 79 14.408 s 14.409 s 14.410 dx 1 cos ax(1 C sin ax) = i 2a(l f sin ax) 14.421 I‘ dx = p sin ax + q(1 + cos ax) q + p tan 5 14.422 dx ax + tan-’ (q/p) psinax+qcosax*~~ 2 14.423 S dx p2 sin2 ax + q2 cos2 ux 14.424 dx = 1 In p tanax - q p2 sin2 ax - q2 COG ax 2apq p tan ax + q sinmP1 ax co@+ l ax m-l - a(m + n) +- sinm-2 ux cosn ax dx I‘ mfn 14.425 sinm uz COP ax dx = sin” + l ax cosnwl ax a(m + n) + n-l m+n s sinm ax COS”-~ ux dx 82 INDEFINITE INTEGRALS 14.459 S dx =x P s dz --- q + p set ax Q Q p + q cos ax 14.460 s se@ ax dx = secne2 ax tan ax n-2 a(n - 1) +- n-1 s se@--2 ax dx ; 1NTEQRALS INVOLVING cm az 14.461 s csc ax dx = k In (csc ax - cot ax) = $ In tan 7 14.462 s cot ax csc2ax dx = -- a 14.463 S csc3 ax dx = - csc CL5 cot c&x 1 UX 2a + z In tan T 14.464 s CSC” ax cot ax dx = _ cscn ax - na dx 14.465 - = -- s cos ax csc r&x a 14.466 .l - x csc ar ,jx = $ f ax + k$ + !k$ + . . . + 2(22n-’ - 1)B,(ax)2n+’ + . . . (2n + 1) ! 14.467 S ?%!!? dx = _ & + $? + !&I?$ + . . . + 2’22’;;n-m1$$;‘2’- ’ + . . . 5 14.460 S x cot ax x csc2 ax dx = - ~ a + $ In sin ax 14.469 S dx = E-I? q + p csc ax Q P S dx p + q sin ax [See 14.3601 14.470 s CSC” ax dx = - CSC~-~ ax cot ax n-2 a(n - 1) +- n-1 S csc”-2 ax dx INTEORALS lNVotVlN@ IRZVRREiZ TR100NQMETRfC fl&CtlONS “’ 14.471 S sin-1 Ed% = U 5 sin-l ZZ + dm a 14.472 ‘xsin-lzdx = sin-l z + X&Z? a 4 14.473 s 39 sin-1 z & = x3 a j- sin-l z + (x2 + 2a2) &K2 9 14.474 S sin-l (x/a) * l l dx = z+- (x/aj3 1 3(x/a)5 1 3 5(x/a)7 + + + . . . 5 2*3*3 2.4.5.5 2*4*6*7*‘7 14.475 14.476 dx - sin-1 (x/u) - $l a-kdG2 = X X 2 - 2x + 2dm sin-l z INDEFINITE INTEGRALS 83 14.477 .(‘ cos-1 :dx = a x cos-1% - @?2 zc,,-l~& = cos-ls _ x a -5 r a 4 14.479 39 cos-l : ,& = i? a 3 cos-1 fj - (x2 + 2a2) &i72 9 14.480 cos-1 (x/a) dx = ;lnx - sin-1 (x/a) x dx x [See 14.4741 14.481 s cos-;;xln) dx = _ cos-1 (x/a) + iln a+~~~ x X > ds = z cos-1 x ( a)2 - 2x - 2dz&os-'~ tan-1Edx = xtan-1E - zIn(xzfa2) a 14.484 x tan-1 Edx = &(x2+ a2) tan-1 x - 7 a x2 tan-1 z dx = 14.486 (x/u)3 (xla)5 tan-~(xiu) dx = ; _ 32 + ~ _ - (x/a)7 72 + *.* 14.487 . 14.488 cot-‘?dx = a x cot-l z + % In (x2 + a2) 14.489 x cot-’ zdx = 4(x” + a2) cot-1 E + 7 52 cot-’ ; dz = ; 14.491 cot-* (x/u) X dx = g In x - tan-’ (x/a) dx X [See 14.4861 14.492 cot-1 (x/a) x2 dx = _ cot-' (x/a) X 14.493 s see-*z dx = ! 2 set-l z - a In (x + &?C3) o<sec-*:<; a x set-* z + a In (x + dm) 5 < set-* 2 < i7 14.494 S x set-1 z dx 2 see-l E - a x-a 7 0 < set-1 z < i = x2 z see-* f + 2 t < set-* t < T x3 ax&F2 14.495 s x2 see-1: ds = i ,secelz - 6 - $In(x + dZ72) 0 < see-1 i < g a X3 ysec-1 z + ax&2G3 6 -t $ln(x+da) i < set-11 < T 84 INDEFINITE INTEGRALS 14.496 .I’ set-l (x/a) dx = ;1nx + ; + w3 + 1~3(cLlX)5 + 1*3*5(a/2)7 + . . . X . . 2-4-5-5 2-4.6-7-7 14.497 s set-l (da) dx = X2 1 _ see-l (x/u) + &GFG _ sec-lx(xiu) &ikS 0 < set-lz < i X ax 5 < set-1 t < T x csc s * csc-1 2 dx = -1: + aIn(x+@=2) 0 < csc-1; < ; 14.498 a xcsc-1: - uln(x+~~) -5 < csc-1 z < 0 X2 2 csc-1 E + a x-a 7 14.499 s x csc-1: dx = 0 < csc-1; < ; a 22 y csc-l % - 2 -5 < ,se-1; < 0 x3 X x2 csc-1 f dx 3 csc-l ; + = X3 X 3 csc-1 a - -5 < csc-1; < 0 * 14.501 s w-1 (x/a) dx = _ E , (dx)3 1 ’ 3(a/x)5 1 ’ 3 l 5(a/x)7 + . . . X X 2-3-3 I 204-5.5 + 2*4*6*7-T _ csc-1 (x/u) - X 14.502 s CSC-~ (x/a) X2 dx = . csc-1 (x/u) - + X 0 < w-1 z < ; ; < csc-1: < 0 xm sin--l 5 dx Xlnfl = ___ a mt1 s 14.505 I' xm tan-1 x dx = a Stan-l: - &Jsdz 14.506 s xm cot-1 f dx = -$+eot-l~ + -&.I'="" xm+l set-l (x/u) mfl 0 < s,1: < 5 xm see-1 z dx = xm+l see-1 (x/a) + a s xm dx - ~ mS1 m+l d= i < set-l% < T xm+l csc-1(x/u) I a ' x"'tzsc-1: dx = i m+l S xm dx m+1 @qr 0 < csc-1 E < ; 14.508 xm+l csc-1 (x/a) mfl -;<cse-$<O INDEFINITE INTEGRALS 14.543 s sinLard = ax I jJ$: / 05 ,. . . . * . 5*5! 87 '14.544 s sinizax dx = * I a x s =Fdx [See 14.5651 14.545 - = S dx sinh ax i In tanh 7 14.546 - = s xdx 1 sinh ax az ax 14.547 s sinhz ax dx = sinh ax cash ax X -- 2a 2 14.548 ,I' x sinha ax dx = x sinh 2ax cash 2ax x2 4a -~-- 8a2 4 14.549 I‘ dx coth ax ~ = sinh2 ax -- a 14.550 .I' sinh ax sinh px dx = sinh (a + p)x sinh (a - p)x %a+p) - aa - P) For a = *p see 14.547. 14.551 ' I sinh ax sin px dx = a cash ax sin px - p sinh ax cos px c&2 + p2 14.552 ' .( sinh ax cos px dx = a cash ax cos px + p sinh ax sin pz a2 + p2 14.553 s dx 1 p + q sinhax = ax+p--m ad~2 qeaz + p + dm > 14.554 s dx - q cash ax +” S dx (p + q sinh ax)2 = a(p2 + q2)(p + q sinh ax) P2 + 92 p + q sinh ax 14.555 S dx p2 + q2 sinh2 ax = 14.556 I‘ dx 1 In p + dm tanh ax p” - q2 sinh2 ax = 2apGP p - dm tanh ax 14.557 S xm sinh ax dx = xrn cash ux m -- xm--l cash ax dx a a I’ [See 14.5851 14.558 ’ sinh” ax dx = sinhn--l ax coshax _ - n-1 S sinhnP2 ax dx an n 14.559 - S sinh ax - sinh ax a cash ax Xn dx = (n _ l)xn-’ + - n-l S QFr dx [See 14.5871 14.560 ~ = S dx - cash ax n-2 S dx -- sinhn ax a(n - 1) sinhnP1 ax 92-l sinh*--2 ax 14.561 ~ = .I’ x dx - x cash ax 1 n-2 S x dx -- ~- sinhn ax a(n - 1) sinhn--l ax - as(n - l)(n - 2) sinhnP2 ax n-l sinhnP2 ax 88 INDEFINITE INTEGRALS INTEGRALS INVOLVING cash ax 14.562 sinh ax cash ax dx = - a 14.563 x cash ax dx = x sinh ax cash ax -- . a a2 14.564 x2 cash ax dz = - 22 cash ax + . a2 14.565 s cash ax (axP -& z lnz+$!!@+@+- X 4*4! 6*6! + . . . * . 14.566 s cos&ax dx = cash ax ; a X s [See 14.5431 14.567 - = s dx cash ax (ad4 + 5(ax)6 + . . . + (-UnE,@42n+2 - - - - 8 144 (2%+2)(272)! + *** 14.569 s cosh2 ax dx = ;+ sinh ax cash ux 2a 14.570 s X2 xcosh2axdz = 4+ x sinh 2ax cash 2ax 4a -8a2 14.571 - = ~ s dx tanh ax cosh2 ax a 14.572 S cash ax cash px dx = sinh (a - p)z + sinh (a + p)x 2(a - P) %a + P) 14.573 s cash ax sin px dx = a sinh ax sin px - p cash ux cos px a2 + p2 14.574 s cash ax cos px dx = a sinh ax cos px + p cash ax sin px a2 + p2 14.575 s dx cash ax + 1 = $tanhy 14.576 s dx = cash ax - 1 -+cothy 14.577 s xdx cash ax + 1 = a !? tanh 7 - -$lncosh f 14.570 x dx cash ax - 1 = --$coth 7 + -$lnsinh 7 14.579 S dx (cash ax + 1)2 = &tanhy - &tanh3y 14.580 s dx = (cash ax - 1)2 & coth 7 - & coths y 14.581 S dx = p + q cash ax tan-’ s ln ( war + p - fi2 qP + p + ) @GF 14.582 s dx q sinh ax P S dx = -- (p + q cash ax)2 a(q2 - p2)(p + q cash as) 42 - P2 p + q coshas INDEFINITE INTEGRALS 89 14.583 s dx = p2 - q2 cosh2 ax I 1 In p tanh ax + dKz 2apllF3 p tanh ax - 14.584 s dx 2wdFW = p2 + q2 cosh2 ax 1 dF2 tan ! 1 In p tanh ax + dn p tanh ax - > dni --1 p tanhax l.h=7 14.585 xm cash ax dx = xm sinh ax _ m s xn--l sinh ax dx a a [See 14.5571 . 14.586 s coshn ax dx = coshn--l ax sinh ax n-1 f- S coshn--2 ax dx an n 14.587 s coshnax dx = -cash ax I a (n - l)xn-1 n-1 s ?$!? ,jx [See 14.5591 sinh ax dx a(n - 1) coshn--l ax coshnPz ax x sinh ax n-2 + (n - l)(n - 2,‘a2 coshn--2 ax ’ - J xdx ~- a(n - 1) coshn--l ax n-l coshn--l: ax INTEGRALS INVOLVCNG sinh ax AND c&t USG .:, ". ' 14.590 ,(' sinh2 ax sinh ax cash ax dx = ~ 2a 14.591 s sinh px cash qx dx = cash (p + q)x + cash (p - q)x 2(P + 9) 2(P - 9) 14.592 s sinhn ax cash ax dx = sinhn + 1 ax (n + 1)a [If n = -1, see 14.615.1 14.593 s coshn ax sinh ax dx = coshn+ l ax (n + 1)a [If n = -1, see 14.604.1 14.594 s sinh 4ax x sinh2 ax cosh2 ax dx = ~ -- 32a 8 14.595 S dx sinh ax cash ax = 1 In tanh ax a 14.596 S dx = _ t tan - 1 sinh ax _ csch ax sinh2 ax cash ax a 14.597 ______ zz - S dx sech a2 sinh ax cosh2 ax a + klntanhy 14.598 dx 2 coth 2ax = - sinh2 ax cosh2 ax a 14.599 S z dx = sinh - i tan-1 sinh ax a 14.600 S ;s,hh2;; dx = cash ax a + ilntanhy 14.601 S dx cash ax (1 + sinh ax)