INTEGRALS AND SERIES

Volume 1

Elementary Functions

By A.P. Prudnikov, Yu. A. Brychkov

and O.I. Marichev

USSR Academy of Sciences Moscow

Translated from the Russian by N.M. Queen

Department of Mathematics The University of Birmingham

England

GORDON AND BREACH SCIENCE PUBLISHERS New York • London • Paris • Montreux • Tokyo • Melbourne

Contents

Preface 21

Translator's Preface 22

Chaplcr I. INDKIIN1TK INTKURALS 23

1.1. Introductlon 23

1.1.1. Preliminary Information 23 1.1.2. Fundamental integrals 23 1.1.3. General fonnulas 24

1.2. The Power and Algebraic Functions 24

1.2.1. Introduction 24 1.2.2. Integrals of the form J*p (oxr + b)q dx '."': 27

1.2.3. Integrals of the form [ * ix 28 * x" La* - r. p xP dx

1.2.4. Integrals of the form \ 5 , 29 J ix + o)*

1.2.5. Integrals of the form \ ~—, 31

1.2.6. Integrals of the form ^ (» + c)p ( j r f ) ' ' dx 32

f x" dx 1.2.7. Integrals of the form } - - g)), - - —r 33 1.2.8. Integrals of the form J Rix. ax' + bx +c)dx 35

1.2.9. Integrals of the form J Rix + d. ax' + bx-l-c) ix 38

1.2.10. Integrals of the form [ x dx . .19 J(*4_toT

1.2.11. Integrals of the form C dx ... 42

1.2.12. Integrals of the form \ ~ ?±- 43 i (x' + o*)"

1.2.13. Integrals of the form V •£- £ - , \ —- i -— d l — 44 J ( t 4 J i i 4 ) " J (« , . | - 6 t 8 + rf'«

1.2.14. Integrals of the form J R U. a*s* + bx'' + c) dt 47

1.2.15. Integrals of the form \ «(x1/2. ax\-b)dx 48

1.2.16. Integrals of the form f fl(Al/J, x"±a") dx **

1.2.17. Integrals of the form J x* "' iax + b)n + ' ' 3 dx 49

f xm dx 1.2.18. Integrals of the forin J (ax + b)" "*"'/J "

f </x 1.2.19. Integrals of the form ) , * (ax + j)« -j-1/-' 52

1

CONTENTS

Integrals of the form J R (x. Vax + b, Ycx + d) dx 52

Integrals of the form J " [x. y'äx + b) dx 55

Integrals of the form J R \x. [ax+b)P:"\ dx 55

Integrals of the form ) " [*• \ — + dJ dx 56

Integrals of the form J R (xMn, x*± tr) dx 57

x Integrals of Ihe form JR(yrx — o. Yx — b. Yx — cldx 57

a 00

Integrals of the form J R (Yx — a, Yx—b. Yx — 1) dx 59 x a

Integrals of the form J R (Ya — x. Yx — b, Yx — c) dx 61 X

X

Integrals of the form J R [Ya — x, \ x — b. Yx — c) dx 6? b h

Integrals of the form j R(Ya — x, Yb — x,Yx — i)dx 64 x x

Integrals of the form J R (Va — x. Yb — x, Yx — r) dx 66 c c

Integrals of the form J R (Ya — x, V b — x. Yc — x) dx 68 -T

x Integrals of the form J R (Ya - x. Yb — x. VT^Tr) dx 70

— CO

x Integrals of the form J « (Yx — a, Yx-b. Yx - c, Yx - d) dx 72

a a

Integrals of the form J R (Ya — x, Yx — b. Yx—t, Yx^d) dx 74 x X

Integrals of the form J R (Ya — x, Yx — b. Yx — c. Yx — d) dx 76 * 6

Integrals of the form J R(Ya— x, Yb — x, Yx — c, Yx — d) dx 78 x x ^_

Integrals of the form \R(Ya — x. Yb — x. Yx—c. Yx — d)dx 80 c c

Integrals of the form J R (Ya—x, Yb — x, Yc — x, Yx — d) dx 82 x x

Integrals of the form f R (Ya —x, Yb — x, Vc — x, Yx — d) dx 84 d d

Integrals of the form \R(Ya—x, Yb—x,\c — x,Yd—x)dx 86 x

Integrals of the form J xm (x- 1_ a"-)n + l / 2 dx 88

CONTENTS

r , , -<.a + 1/2 ntegrals of the form \ \"~ i " ' j x

J xm

ntegrals of the form C * il

ntegrals of ine form V ~ :—

ntegrals of the form \ \ . . . .

J (x + fr)" Vx1 i a> J (x« ± fr=) 1 x> i « ' ntegrals of the form J x m (a" — x 2 ) " ' <fx

S tfli _ 1*»" + ' ^ ; - ; — — - (/x

r \ m tir ntegrals of the form \ i—i-1—_

ntegrals of the form \ — ;—

ntegrals of the form f * • _ _ _ , V — —

J ( x " ± * " ) " V ? r ? J (x> + 4 !) (as - x - ) " + >l-

ntegrals of the form \ x~'-m (ox' + frv-|-on -*- ''"•* djc

f x'~ In dx ntegrals of the form \ —

> (a.vI + frx + £) ' ' + ' ' '

ntegrals of the form J / * ( * + />, ox ! + frx + c)dx

ntegrals of the form J / (x, fx' — x + l) dx

x

ntegrals of the form J R (x. Yx' + a*. Yx> + fr=) dx 0 <»

ntegrals of the form f R(x. y'x' + a'. \'j? + fr*) dx x x

ntegrals of the form [ R (.«, > 'V + a'. Yx' — b') dx 6 o°

ntegrals of the form J R (*. Yx- + a\ Yx' — 6-) </x x X

ntegrals of the form / ß (>'• *'*' + fl:. l'fr' -.v«) dt i) fr

ntegrals of the form I * (*• » ' , ' l " : - > V - . v ) </x

x ntegrah of the form [ R (x . / x » —o». y V — fr3) <fx

a oo

ntegrals of the form J fl (x, / A ! - O ' , YX- — b2) dx x a

ntegrals of the form J R (x. Ya' — x*. Yx' — b>) dx x X

ntegrals of the form J R U. Ya' — x'. Yx' — b') dx fr

89

91

92

93

95

96

97

98

99

100

102

KM

106

107

109

110

112

113

114

116

117

118

119

4 CONTENTS

X

1.2.65. Integrals of the form J R (x. V'a' — x', VB1 — X>) dx 121 u b

1.2.66. Integrals of the form i J Rix. Ya' — x', Yb' — x*) dx 122 x

1.2.67. Integrals of the form J R(". V{X' + p=) u' - f p')rf.i 123

1.2.68. Integrals of the form J R (x, Yx' + l) dx 124

1.2.69. Integrals of the form J R ix. Yx' — l) dx 125

1.2.70. Integrals of the form J Rix. Y~\ — x') dx 126

1.2.71. Integrals of the form J R(x. Yx' + \) dx 128

1.2.72. Integrals of the form J Rix, Yx ? -F 0 dx 129

1.2.73. Integrals of the form / I ix, Yx + x') dx 130

1.2.74. Integrals of the form J Rix. Yx — x') dx 130

1.2.75. Integrals of the form J R ix. Yx> + ü&'.i* + o<) dx 131

1.2.76. Integrals of the form J R 'x. Yx'+ l) dx 132

1.2.77. Integrals of the form J Rix, / l — . v ) dx 133

1.2.78. Integrals of the form J R(x, y'x'S l) dx 134

1.2.79. Integrals of the form J R(x. (,' x> ± l) dx. J /?(r. y'T^T') dx 134

1.2.80. Integrals of the form J R(yH ix — o), ty {x — b)) dx 136

1.2.81. Integrals of the form j £ ( | / " * 4 + ') dx 136

1.3. The Exponentlal Function 136

1.3.1. Integrals of the form / / W") dx 136

1.3.2. Integrals of the form J / ix, cax) dx 137

1.3.3. Integrals of the form J / U. t~a'x')dx 139

1.4. Hyperbolic Functions 141

1.4.1. Introduction '141

CONTENTS 5

1.4.2. Integrals of the form J shp x äx, J ch" x äx 141

1.4.3. Integrals of the form J shp x ch? x dx 142

1.4.4. Integrals of the form \ — ~ äx. \ c ' „ * äx 144 J ch? x •> =hp x

1.4.5. Integrals of the form f rf,t - 14(.

1.4.6. Integrals of the form / th p x dx, J cthp x äx , 147

1.4.7. Integrals of the form f R (sh .r, ch x, th x, cth x) dx 147

1.4.8. Integrals of the form J R{sh(ax + b). ch (ex + </)) dx 151

1.4.9. Integrals of the form \ sh" x {** "*} dx, $ ^ •* {ch o«} ^ 1 5 1

1.4.10. Integrals of the form J R (sh lax, ch Tax, V"sh lax) dx 153

1.4.11. Integrals of the form J R(sh!ax, chiax. \rciTiäx) äx 154

1.4.12. Integrals of the form f R(sh x, ch x, Ya + b sh x) d t 155

1.4.13. Integrals of the form J R(shx, e h r , V<i ch x - n ) dx, b>a 156

1.4.14. Integrals of the form J /?(shx, ch jr, V* ch x — a) äx, a > b 156

1.4.15. Integrals of the form J R (sh x, ch x, Yo~bchx) dx 157

1.4.16. Integrals of the form / R (sh x, ch x. Ya + b ch x) dx 158

1.4.17. Integrals of the form J R(shx, eh.», V a> sh !x±6- ' , /»» —a J sh 'x , 159

Va'ch'x±b', V*» —a'ch'x) dx

1.4.18. Integrals of the form J R (sh x, ch x. / a sh x + b ch x) dx 1«)

1.4.19. Integrals of the form f T'tFxdx, J VcThxVx 160

1.4.20. Integrals of the form $ x" /*jj *1 dx 161

$ 1 fsh*)?

~P Ich xj ** I 6 2

1.4.22. Integrals of the form \ xp \ ^ x \ dx 163

1.4.23. Integrals of the form \ *? shp x ch» x äx 164

1.4.24. Integrals of the form ( - i — äx, \ —— äx 164 J s h ? x -»ch^x

1.4.25. Integrals of the form \ R {x?, {*£*}. a + b {*£ *}) dx 166 1.4.26. Integrals of the form $ ibx + c)±" {*h°*} dx 166

1.4.27. Integrals or the form J R(x, eax, sh ix, ch bx) äx 167

1.5. Trigonometrie Functions 168

1.5.1. Introduction 168

1.5.2. Integrals of the form f s ln p xdx 169

1.5.3. Integrals of the form J cos"xdx 170

6 CONTENTS

1.5.4. Integrals of Ihe form J sinp x cosf x ix 172

1.5.5. Integrals of the form \ ix 174 J COi? X

1.5.6. Integrals of the form \ — ix 176 J sin^ x

1.5.7. Integrals of the form \ —-—•——— 178 •> sin^ x cos* x

1.5.8. Integrals of the form J tu''x äx, J cls''xix 179

1.5.9. Integrals ot the form J Ä(sinx, cos*, lg JT. ctg x) ix 180

1.5.10. Integrals of the form ( | S i " <" + f s i n <" + "] \ ix. 185 " .' leos (ax-i-b) cos [cx + i)l

J sin {ax + b) cos {ex + i) ix

1.5.11. Integrals of the form J i\np x %\n ax ix 185

1.5.12. Integrals of the form f sin'' x cos ax ix 186

1.5.13. Integrals of the form J cosp x sin ax ix 187

1.5.14. Integrals of the form J cosp .r cos ax ix 188

1.5.15. Integrals of the form U s i " v l " / s i n "")' ix 189 b J leos x) (cos nx)

1.5.16. Integrals of the form f R (sin ax, cos ax, / s i n 2m) Je 190

1.5.17. Integrals of the form J R (sin o.r, cos ax. V cos lax) ix 192

1.5.18. Integrals of the form J fl(sina.r, cosa.r, V— cos '.'a v) ix 194

1.5.19. Integrals of the form j R(s.\nx, cos.r, Ya ±b sin x) ix 195

1.5.20. Integrals of the form J R (sin x, cos x, Ya~±~bcöTx, ix 197

1.5.21. Integrals of the form J R (sin x, cos x. Ya + b sin x + c cos x) ix 199

1.5.22. Integrals of the form J R (l'l — k' sin'.t) äx 200

1.5.23. Integrals of the form J R (sin x, Y\ —k' sin-' x) ix 200

1.5.24. Integrals of the form J «(cos.r, Y\—k' »in« x) ix 202

1.5.25. Integrals of the form \ sinm .t cos" JC }' (l — k1 siir .t)'' dx 204

1.5.26. Integrals of the form \ UL-i. y | _ k' si »' x ix 206 •> cos" .t

1.5.27. Integrals of the form \ _ Ü ^ _ £ £ ° i _ £ _ ix 207 V (1 — k! sin'.v)''

i e -io i . i t .x. c f sin'1 .t rf.r r cosp .r d.t - , , M » 1.5.28. Integrals of the form \ . , \ - 209

cos't V (I — k' sin* xf Jsin'?.v] (I — i* sin" xf

1.5.29. Integrals of the form \ ^~k'Si"'x^ ix 211 J sin jrcos x

1.5.30. Integrals of the form \ ' 211 J si nm x cos" x V (I — k'2 si nä x)r

1.5.31. Integrals of the form \ " ± ^ I ^ ' L ' ^ J i 212 •> V'l—ft'sin'.r

1.5.32. Integrals of the form f (a->-cos t ) p . 213 \ —. •=. ix •> KI —A'sin-.v

CONTENTS 7

1.5.33. Integrals of the form f (a-f \?.r)p 214 \ = •=. dx J V I— fc'sin'jt

1.5.34. Integrals of the form J fl(»lnjr, coä.t, yi—p'liüTJt, Vl — ii'iin-x) dx 216

1.5.35. Integrals of the form [ ( a + ? "" f ) dx 217

•> V l — k'i\n'x

1.5.36. Integrals of the form J /?(sin.v, cos.r, V\ — p-sUt:x) dx 218

1.5.37. Integrals of the form J R(sinjr, cos.r, V\+p'-sm'x) dx 219

1.5.38. Integrals of the form /«(s in . r , cos.r. Vu'-iM'x- \) dx 220

1.5.39. Integrals of the form J /de*)dx,.[ I(ctgx)dx 221

1.5.40. Integrals of the form ^ x" {*',£*}* dx 223

1.5.41. Integrals of the form \ -\ l s i " M 7 dx 225

•> xp Uos.vl

1.5.42. Integrals of the form [ xp I ]?*.)' •*•* 226

1.5.43. Integrals of the form J .tr W . v co»'* dx 227

1.5.44. Integrals of the form \ —£l 228

1.5.45. Integrals of the form ? ' dx 229 J cos7 x

1.5.46. Integrals of the form [Rix'1, sin.r. cos.r. a-V fcsinA -fr coix) dx 230

C vcinm reo5"*:

1.5.47. Integrals of the form \ -','" I "—— dx 231

•U / ( l - IS i i in ! . r ) ' -

1.5.48. Integrals of the form \ (x + tY- " {-inax\ dx 231

1.5.49. Integrals of the form J c u J i : i ' i t o ; 1 xdx 232 1.5.50. Integrals of the form \ «OA'{1**V' dx 234

1.5.51. Integrals of the form jR(.r. tax, sint.t. cost.v)i/t 234

1.5.52. Integrals of the form \ { ^ t ^ CZVdT " 235

1.5.53. Integrals of the form [ x" \imx*\ dx 240

1.6. The Logarithmic Function 240

1.6.1. Integrals of the form \xrh,"xdx 240 p .

1.6.2. Integrals of the form \ x 242 •> ln''.t

1.6.3. Integrals of the form J U + a i ' inxdx 242

1.6.4. Integrals of the form \ x ' ' ' * ; dx 243

1.6.5. Integrals of the form J xp l:\(ax + b)dx 243

1.6.6. Integrals of the form \ v±" ; In j^TZ dx 245

1.6.7. Integrals of the form J * - '" In U ' ± o " ) dx 245

1.6.8. Integrals of the form J t i m In (v + V x-±a-)dx 247

8 CONTENTS

1.6.9. Integrals of the form \ *' In (v + Vx- ± l) dx. [ l,v (x + Vxi± l) äx 248 •I V ,<; .i. I •'

1.6.10. Integrals of the form \i(x, li.v, eax, «int, ccs.v. ...)dx 249

1.6.11. Integrals of the form \ t^l±^l±Jl j , 251

1.7. Inverse Trigonometrie Functions 253

1.7.1. Introduction 253 1.7.2. Integrals of the form \ Z"'«'"'« "M" dx 253

1.7.3. Integrals of the form [ x±n farcsin x\ dx 254 J larccosjc)

1.7.4. Integrals of the form f ( l H ^ , ± n + l / 2 l""'in*\dx 2 5 5

J larccos*/

1.7.5. Integrals of the form [ x" (l-x2)" + l/2i""inx\rdie 256 J larccosjtj

1.7.6. Integrals of the form [ x" ( a r c s e c (*'a) ,1 dx 259

1.7.7. Integrals of the form [ (r- i - l )±" + l/2 {a r c s e c J ! \dx 261 J "~ larecosee x)

1.7.8. Integrals of the form [ xm (x--l)±a + l'2 l""1*' } dx 262

1.7.9. Integrals of the form \ x» { ' ^ j ^ , } <" 264

1.7.10. Integrals of the form \ *p (*2 + S)« { ^ [ Z } ' <" 266

1.8. Inverse Hyperbolic Functions 268

Chapler 2. DEFINITE INTEGRALS 270

2.1. Introduction 270

2.1.1. Preliminary Information 270 2.1.2. General formulas 271

2.2. The Power and Algebraic Functions 279

2.2.1. Introduction 279

2.2.2. Integrals of general form 279 2.2.3. Integrals of (^To*1)'' and ( o f - x " P 295 2.2.4. Integrals of x

a (a" ±.t")p 296 2.2.5. Integrals of (of + ^ ) P ( 6 v + * v ) f f 298

3 2.2.6. Integrals of J J (ak±xfk 301

* = l 3 ,

2.2.7. Integrals of TT f a > * * ± , l » * W 304

n 2.2.8. Integrals of TT « ^ ± r " * ) p » , n > 4 306

2.2.9. Integrals of *° (axi+bx+c)(> A{x) 308

CONTENTS 9

2.2.10. Integrals o f» a (a* '+26« 2 +c) ' ) 313 2.2.11. Integrals containing A (V^ax' + Ox + c) 313 2.2.12. Integrals containing differences of algebraic functions 317

2.3. The Exponential Function 318

2.3.1. Introduction 318 2.3.2. Integrals of general form 318

2.3.3. Integrals of . r V P * 322

2.3.4. Integrals of (x±afe'l'x 322

2.3.5. Integrals of (S-±an)Pr?* 323

2.3.6. Integrals of x" (x^afe'l"1 324

2.3.7. Integralsof x<t (x*,-azy>c-px and *a(n5—x-) !'e~p x • • • 326

2.3.8. Integrals of , « J | ( a ^ * ± x > l * ) ' ) f t f * 328 k

2.3.9. Integrals of , a ( / r + 7 + a x H + j , u ) P f - p j r 329 2.3.10. Integrals of xa(irxT+r,+ax + bl)Pe-px 329

2.3.11 Integrals of , * [(V^FT"b±V7xT+äf ± WTx^TbT V^TdYYP* •••• 330 2.3.12. Integrals of xa(t('x+ zYrpx 333

2.3.13. Integrals of xa{t1x_j)VJr 337

2.3.14. Integrals of {at-1x + bt(>x+ c)~,Hx) 341

2.3.15. Integrals of Aix)e-Px'—"x 343 2.3.16. Integrals of A(x)e-Px-1/* 344

2.3.17. Integrals of AMe-'>^x'^-a'~Px 345

2.3.18. Integrals of ig'x'l(x) 346 2.3.19. Integrals containing differences of algebraic functions and the exponential

function 347

2.4. Hyperbolic Functions 349

2.4.1. Introduction 349

2.4.2. Integrals of general form 350

2.4.3. Integrals of >l (*){*„ **}° 351

2.4.4. Integrals of TT ( s h V \ . 353

l l \ c h V ! 2.4.5. Integrals of A{x)J\ f* ***} 354

2.4.6. Integrals of (fl + 6ch''6x + csh' ,^)'>TT (* '"*) 356

V lch v J 2.4.7. Integralsof AM(a + bcUnbx+c<>hnbx)<>T\ i ^ V ] 359

2.4.8. Integrals of /»(-Ochfcx { s h^"'-*'>. 360 Ichl-V—A'I

2.4.9. Integrals containing th ax, cth ax and |the differences of hyperbolic and algebraic functions 361

2.4.10. Integrals of *<Vpj t{*Ji **|° 361

2.4.11. Integrals of e_ /" and hyperbolic functions 363 2.4.12. Integrals of AU), e1" and hyperbolic functions 364

10 CONTENTS

2.4.13. Integrals of xa(ex+a)prpxl*h bx\ 365

Ich bx) 2.4.14. Integrals of x < V f { b + s h a x ] \ " 366

2.4.15. Integrals of xae~ax'-" Js l ' **l 367 ° Ich&xJ

2.4.16. Integralsof AMexp(-pVrx~i+7') f!""! 369 Ich ax)

2.4.17. Integrals of * V " |*Jj **} 369

2.4.18. Integrals of /(x)exp {*£ **} 370

2.4.19. Integralsof y i u > « - ^ { $ h r " ' + " x + c \ 371

2.4.2Ü. integralsof A,*>e><*> { * j * ^ ) 373

2.4.21. Integrals containing /l(x). </'-v\ sha(x), ch<Mx) 374

2.4.22. Integrals containing differences of A(x), e'P*, / s h b*\ 375 Ich bx)

2.5. Trigonometrie Functions 378

2.5.1. Introduction 378 2.5.2. Integrals of general forin 378

2.5.3. Integrals of x a ( s i n bx\" 386 6 leosox)

2.5.4. Integrals of xm / s i n ' l - " 388

2.5.5. Integrals of (x + j f {^**1 389

2.5.6. Integrals of U » ± . » ^ } " . < - - ^ { £ £ } " 389 2.5.7. Integralsof x » U i ^ { * J 3 x * ( a - x , > ' { £ £ } 39,

2.5.8. Integrals of x«(x'- a*}M { £ £ } . x«(*'-W { £ £ } 393

2.5.9. Integrals of fü J s i n bx\n 394

xm <i\nbx\l 2.5.10. Integrals of „ ^ n + % _ i X « - i + „ . + a § W » x J 3 %

2.5.11. Integralsof i a ( l AI

: + 2 ! t « + ^ ) V ^ U ) [*'0" "x] 399

n |sinfl1,xl>lAfsin(i.jr1

v(, -J-J „ v

ft ( „ . y («,6,4 • n «»"»v». **** 400 2.5.13. Integrals of x°sin , ,axcosv6x 404

x a

2.5.14. Integrals of x , - , sin" ax cos' bx 409 2.5.15. Integrals of x ° J J s i n M n^xeos ' ^x 411

k

2.5.16. Integrals containing f a + fc{sin " 1 ) P 412

2.5.17. Integrals containing (ocosx + frsinx + c)" 420 2.5.18. Integrals containing (acos5x4-6sinäx + c)~" 422

2.5.19. Integrals containing (acos s x4- ts in 5 x+c) 425

CONTENTS 11

2.5.20. Integrals of A«) JSi" " } " {S i n ( ,f + f : ) V . X W [s in « * } " f 8 ! " ' * ' * ' ? } * 429 " UOSOJCJ lcos(öx-(-c)J IcosaxJ lsiu((ix4-c) 1

2.5.21. Integrals ol l \ „ T „ \ r 429 \cos(axl' + bx1+c)\

2.5.22. Integrals of x<* J s " " - ^ s ' « "'), x « f«n ox? cos 6x1 4 3 0

Uosaxpcos 6xJ Icosax^ sin 6xJ

2.5.23. Integrals of JH*) f » M « + */*)! 432 '\cos(ax-)-6/x)l •

2.5.24. Integrals of *« (""<«/*> ««»«1 x « f*n (a/x) cos 6x. 4 3 3 6 lcos(a/x)cos6xr \cos(a/x)sin6xJ

- . , , , , , . . . . h\n(cYa' + x*) sin6x\ (sin (cv'o=Tx=)cos6.xl . , . 2.5.25. Integrals ol .4(x) { , , _ r . >, A (x) • ; . - > 434

leos (cVa2^;xvcos6xJ lcos(cl a s : ix a l sin6xj 2.5.2(i. Integrals containing 1 c l „ a c f 436

2.5.27. Integrals containing trigonomelric functions of trigonometric functions 438

2.5.28. Integrals containing x" and trigonometric functions of trigonometric functions 440 2.5.29. Integrals containing differences of trigonometric and algebraie functions 441

2.5.30. Integrals of ,~P* J5 i n * * \ " 444 (cos6xJ

2.5.31. Integrals of . V f " b'\ 446 leos 6t-J

2.5.32. Integrals of A(x)rPx ( s i n bx\ 447 leos fcxj

2.5.33. Integrals of x V » f " "*]» f n * x } v ' . A ^ r f axeofbx 448 " IcosaxJ \cos6xJ

2.5.34. Integrals of xaR Uax) | S i n bx\ 449

* leos bxj 2.5.35. Integrals of / (x, eax. sin6x. cosfcx) 451 2.5.36. Integrals containing ,— ax' — ex fsin >>*\ 451

leos 6xJ

2.5.37. Integrals of x<*-,-px-qlx fsln bx\ 453 leosoxj

2.5.38. Integrals containing e -o /x> ea Yx / s i» bx\ 454 \cos6xJ

2.5.39. Integrals containing eA<" and e^(x) „ {*'" **l 455

2.5.40. Integrals containing the exponential funciion of the exponential and trigono­

melric functions and ism bx\ 456 leos6x1

2.5.41. Integrals containing -4(x). J <*>. i s i n ( < " " + * * ± l + c \ 457 (.cos (a*- + 6x± '+<•))

-, c ,.-> 1 . 1 r , . < - P t (sinT'ax»-J-6x-firl 2.5.42. Integrals of A{.x)e r* 1 > 459

lcosVax*-f- bx-j-c)

2.5.43:' Integrals of ,,x,«Mx» { ^ X l ^ } 462

2.5.44. Integrals containing the exponential and trigonometric functions of trigonomelric functions 463 2.5.45. Integrals containing differences of the functions A(x), e~?x, il'n \ 465

leos 6x1

r -w-r fsh a.x\^i, fsin 6,,X|V* -i—r fsl> ". x\>Li, fcos6,x}v» 2.5.46. Integrals o f T T J * V < * > T T < * > < M 467

J. 1 \ch a^xj \cos6AxJ • l l ^ c h o ^ x j \sin 6^xJ „ ™ T-r fsn "tx"!11* (sin 6 t x» v * „-i—r |sh «.xl^k i c o s i . x i v t

2.5.47. Integrals of xa T\ , M J * \ . xa T T J u * l J . * l 469 l l . | c h a / ( x j ^cosft^xl l l l c h a ^ x j \sin6AxJ

2.5 48. Integrals containing yf(x)(a + 6 c h x ) v i 5 1 n " 1 470

12 CONTENTS

2.5.49. Integrals containing A(x)(a-j-*ch»x)v {*'"_ex\ 472

2.5.50. Integrals containing A (*) <chax± cos bx)-' { '^""j 472

. . . _ * . . fsh axl~» rsin bx\ 2.5.51. Integrals containing I* + * ' ~ , { c h ax] \cosbx) 473

2.5.52. Integrals containing A (x), sh bx, ch bx, sin ax', cos bx' 476

2.5.53. Integrals of A <x) i s h ^ S E * ? , s i " H -4 <x>/sh l ' j E Z ! C ° S " Xf 477

Ich (Vfl» — x') cos bx) (ch (P a' — x») sin 6x) 2.5.54. Integrals containing trigonometric functions of hyperbolic functions 478 2.5.55. Integrals containing hyperbolic functions of trigonometric functions 479 2.5.56. Integrals containing „- »>•*, sh ax, ch ax, sin (>x. cos &x 480

2.5.57. Integrals containing e*"". sh ax. ch ax, sin bx, cos bx 481

2.6. The Logarithmic Function 483

2.6.1. Introduction 483 2.6.2. Integrals of general form 483

2.6.3. Integrals of xa In0 x 488

2.6.4. Integrals of • - * — lnffx 488

x° 2.6.5 Integrals of lnffx 490

(o11— x>')v

2.6.6. Integrals of * ? ' " " * , 492 ax'+bx + c

2.6.7. Integrals of x a ( a s - x 2 ) " In" x 495 2.6.8. Integrals of A(x) ln"x 496 2.6.9. Integrals of xa ln"(ar + o) 498 2.6.10. Integralsof xa(ax + bft In" icx + d) 499 2.6.11. Integralsof x a (ax* + fc)P ln(fx4-rf) 506 2.6.12. Integrals of *a(aix4-6I)P(o»v + *,)vIn"<« + <>) 507 2.6.13. Integrals of x a In" ^ - ± J 510 2.6.14. Integralsof A(x) ln(ax'-f 6x+r) 512 2.6.15. Integrals of Alx) In £-jjÜ 514

2.6.16. Integrals of -4(x)In° Wax» +b^ + cxv) 515

2.6.17. Integrals containing A(x) and ! 520 ln"x

2.6.18. Integrals of A W ' " " * 524 ln m x + a

2.6.19. Integrals of products of logarithms 525 2.6.20. Integrals containing In In x 527

2.6.21. Integrals of x V j r f l In"x 527

2.6.22. Integrals of x a e ax l + 6x±' | n " x 528

2.6.23. Integrals of xat'px In (a + 6x) 529

2.6.24. Integralsof A (x) e~px In (ax2 + bx + c) 530

2.6.25. Integrals of e~pxlnA (x) 531 2.6.26. Integrals of « ( « * ) " ix 532

2.6.27. Integrals containing x°. e~r" and ! 532 ln"(x + a)

CONTENTS 13

2.6.28. Integrals of x" ln(a + be~l"c) 533 2.6.29. Integrals of x°R(shx, chx) Inx 533 2.6.30. Integrals of A Ix) R (shx, chx) In A(x) 534

2.6.31. Integrals containing the logarithmic function of hyperbolic functions 535'

2.6.32. Integrals of A <*){*££} In» , 536

2.6.33. Integrals containing A <*)( , i n**) \nAtx) 537

2.6.34. Integrals containing In [5,n bx\ 539

2.6.35. Integrals containing In" (*'0"£*} 543,

2.6.36. Integrals of A (x, sinx, cosx) h i tncos*+ *slnx + c) 544

2.6.37. Integrals containing In-! ' , ±—l 546

2.6.38. Integrals containing ln(o cos'x + *sin'x + c) 547 2.6.39. Integrals containing In A (tgx) 549 2.6.40. Integrals containing the exponential, logarithmic, hyperbolic, and irigonometric functions 55c

2.7. Inverse Trigonometrie Functions 552

2.7.1. Introduction 552J 2.7.2. Integrals of general form 552

2.7.3. Integrals containing farcsin »*l 5 5 6

larecosixj 2.7.4. Integralsof A ix). { " £ * * , } " 557

2.7.5. Integrals of » <•»>• '~P*. {"***,} 559

2.7.6. l ^ o ( , W . p t a 560

2.7.7. Integrals containing arctg ax and the logarithmic function 561

2.7.8. Integrals containing / •««* / (x ) 1 5 6 |

\arcctg/(x) l 2.7.9. Integrals containing / ( { J ™ ^ } ) 562

2.8. Inverse Hyperbolic Functions 563

Chapter 3. MULTIPLE INTEGRALS 564

3.1. Double Integrals 564

3.1.1. Introduction 564 3.1.2. General formulas; integrals of algebraic functions 565 3.1.3. Integrals containing the exponential function 567 3.1.4. Integrals containing hyperbolic functions 572 3.1.5. Integrals containing trigonometric functions 573 3.1.6. Integrals containing the logarithmic function . 579 3.1.7. Integrals containing inverse Irigonometric and inverse hyperbolic functions 580

3.2. Triple Integrals 580

3.2.1. Introduction 580 3.2.2. General formulas; integrals of algebraic functions 583 3.2.3. Integrals containing the exponential, hyperbolic, and trigonometric functions 584

14 CONTENTS

3.3. Multiple Integrals 585

3.3.1. Introduction 585

3.3.2. Integrals over the region AJ + JLJ+ ... + A ^ : S / " 2 585

, «1 x" i * « -

3.3.3. Integrals over the region 1 (-... + —— < I 588 "l "i n

3.3.4. Integrals over the region x ^ o, x„ > 0 xn 2* 0, *, + *, + ••• + *n < a 590

3.3.5. Integrals over the region r \tix\dx 593

a a

Chapter 4. FINITE SUMS 596

4.1. Introduction 5 %

4.1.1. Sums of the form £ ( ± 0 * ^ (*) 596

V ( i ' l )* 4.1.2. Sums of the form Jj ~}— 599

V ('•!)* 4.1.3. Sums of the form > , " . 600

*••• (s -̂ a)

4.1.4. Sums of the form J] ik + auk + 'b) * • • • • • 600

4

4.1.5. Sumsoftheform ^ - g Ü i - _ 60.

4.1.6. Süms of the form V pr̂ —- and others 602

4.1.7. Sums of the form ^ " j * * 603

4.1.8. Sums of the form S\ (±1)* [—1 605

4.2. Binomial Coefficients . . 606

4.2.1. Sums of the form 2 <±'>* L * j 606

4.2.2. Sums of the form V a J * ) 608

4.2.3. Sums of the form V „ (**\x* 612

4.2.4. Sums of the form V 1 a [ * ) 615

4.2.5. Sums of the form V (-H)* l ' ° * U M 616

^ \ V Vkl 4.2.6. Sums of the form y < a (bk\ / *M 622

4.2.7. Sums of the form y i fl /** \ / M j * 624

4.2.8. Sums of the form v 1 - / M / M ~ ' 628

^ *WUJ 2.9. Sumsoftheform £ « , ( £ ) ( £ ) ( £ )

4.2.10. Sums of the form £ ak [**] / M ( '* ] " '

CONTENTS 15

4.3 Hyperbolic Functions 635

4.3.1. Sumsof the form V ak {s!l '** t "\\ 635

4.3.2. Sums containing sech J: and th x 636

4.4. Trigonometrie Functions 637

y i rsin(fejr + a)l 4.4.1. Sums of the form Z-i * lco5(fci + o ) / 637

y i fsinftJtlm

4.4.2. Sums of the form 2-1 °* tcos**/ 639

4.4.3. Sums of the form Zu * \co3*4xj 641

V< fsinfcxl' <s\nky\m

4.4.4. Sumsof the form 2 J ° * \ C O S * * J \cos*j , / M l

4.4.5. Various sums containing sin x and cos x 642 4.4.6. Sums containing sec x and cosec x 644 4.4.7. Sums containing tg ak and ctg ak 646 4.4.8. Sums containing logarithms 647 4.4.9. Sums containing aretangents 649

Chapter 5. SERIES 650

5.1. Numerical Series 650

5.1.1. Introduclion 650

y (±ir • " (ft + oC

5.1.4. Series of the form V ( J : I ) * 653 ^-J (2*-f l ) s

5.1.5. Series of the form V 1 <—"* 654

5.1.2. Series of the form V ( ± l > * 651 ks

5.1.3. Series of the form V <±'>* 652

Z-l kn +

Scries of the form V^ ( IM* 5.1.6. Series of the form V < - " 655

5.1.7. Series of the form \ * . , ~ ,— Z-l k{kn + n

•sri (> 11" 5.1.8. Series of the form 2 j {k + l)(kn + m)

y i (11)* 5.1.9. Series of the form zLi (2fc+ /)(*"

z

l (5fe + /) (ftn + m)

5.1.13. Series of the form V <±D

656

657

P ^ T 659

(±1) 5.1.10. Series of the form Z-l {Zk +1) ikn + m) 660

,* ( £ 1 ) ' /)(*"

5.1.12. Series of the form y 1 ( I D * 663 2-1 i

5.1.11. Series of the form Z-l (4k + /) (fcrt + " ' ) ^ '

V . (£!>- 663 Z J (6* + /) (*n + m)

5.1.14. Series of the form y i (±1)* 664 ^-1 (8ft + /) ikn 4- m)

16 CONTENTS

5.1 .15 . Series of the form V LLÜ 665 Z-1 (An + b) {ka + c) {ka + d)

5.1.16. Series of the form V — ( i ' ' . — - 666 Z J k {kn, + m,| {kn, + m,)

5.1 .17. Series of the form V » <+.'>* 668 Z-i {k+l){kn, + m,)(*n,+ «i,)

5 .1 .18 . Series of the form V Lki> 670 Z j (2* + m) (Art, + m,) (An, + m j

5 .1 .19 . Series of the form V — 6 7 3

Zj(3A + ,* 5.1.20. Series of the form V — — — - , ' " . .... , - , Z J (-1* + m)C<* + ra,)(« +m,)

(2* -f m) (Art, + m,) (An, + mj

m) (3*+riö|3*+ffl i )

<J^>* 674 i i )

5 .1 .21 . Series of the form 2 (An, - f m,) (An, + m.) (An, + »«,) (An, + m j 6 7 4

5.1 .22 . Series of the form V _^ 4 ' ' : 6 7 9

^_l (An, + m t) (An, + m.) ... (An, + in,)

5.1.23. Series of the form I ] (to1 + « l ) ( t n 1 + ' ^ ) . . . ( t a < + BI<)- ' = 6 ' 7- 8- 9 6 8 °

(II) * 5.1.24. Series of the form J J > + fl p r j r ^ TITT^ ; 6 8 2 :l)(*+«2) - ( * + " « )

( l i l ' i i »

,*

685 5 .1 .25 . Series of the form V ( - f^ ^>

5.1.26. Seriesof the form V » n* ( 2»„+ " ; " ' 688 *- ' [(2*+ I)-Jra-Jn

5.1.27. Series of the form V - 1 H L 691 ZJ ( Ä 4 i a »)m

5.1 .28 . Series of the form V LLL> 692 Z- J l(2*+ l) '±o4Jm

5.1.29. Series of the form V (, """* 693 Z J " , t a M

5.1.30. Series of the form V "-± 694

5.1.31. Series of the form V "j 694

//V (A) \ 5 .1 .32 . Series of the form V a ^ V bm) 695

5.2. Power Series 695

5 .2 .1 . Inlroduction 695

5.2.2. Series of the form Y P (A) i* , • • • • 696

(A + o)s 6lJ7

5.2.4. Series of the form V (~}] x* 697 ZJ nk + m

5.2.5. Series of the form 2 j („A + m) l/A + / ) ' 6 " h

5.2.6. Series of the form V LL1L_£ ;>•> 702 Z J (An, + m j (Ana + m2) ... ( * « , + «!,•)• ' "

CONTENTS 17

5.2.7. Series of the form y ( 1 0 * x* y (,tQ*** 703 Z j ( / i * + m)l" ZjT{ka-t-b)

5.2.8. Series of the form y 1k xk 705 ^- J (n* + m)! (ft + a) '

5.2.9. Series of the form y k" j , 706 Z J I* +m)l '

5.2.10. Series of the form V aJi ,* 707 Z j r (*a + b) V ykc + ä)

5.2.11. Series of the form V * -— x* 709 £—1 r {kc -f- a)

5.2.12. Series of the form V * „ ———<- xk 711 Z-l r (da + b) V (iw + d) T (/« +1)

5.2.13. Series of the form £ , . , > / + d , r<*, + /, ' * 7.1

5.2.14. Series of the form £ " / ' * V ; £ X + * ** 7 1 3

5.2.15. Series of the form 2 j "* r i t e + / i n » | + *l 7 I 4

v „ r ( * > i + ' i ) - r ( * t < - t - f i ) j > 5.2.16. Series of the form ZjTlkd +eJ)...VIkd +e ) 714

2 1 V" <£•>' \ ft

"* ( 2 J «/ + m I * 715

5.2.18. Various series of the form S " » * * 717

5.3. Series Involving the Exponential and HyperboUc Functions 718

5.3.1. Series of the form y a ' 718 Z j * j n * n ± l

5.3.2. Series of the form 2 <y7<*a + *•* 719

5.3.3. Series containing sh akx and ch akx 720

5.3.4. Series of the form V a cosech nx 721

5.3.5. Series of the form V „ cosech kn 721

5.3.6. Series of the form V a Eech 6 c 722 ^J n n

5.3.7. Series of the form V a-sechft.n 723

5.3.8. Series of the form V 0fc thA^ 724

5.3.9. Series of the form V ak eth kn 725

5.4. Trigonometrie Series 725

5.4.1. Introduction 725

5.4.2. Series of the form y ( t D 1 ' fsln**l 7 2 6

Z j ks \coä kx I

5.4.3. Series of the form V ,<-'•"* («'"<»*-r >) 1 727 Z J (fc-f a)s \cos(tx+6)i

5.4.4. Series of the form V <:")* fsintxl 7 2 y Z J (*<it+mi)(*n, + mf) IcosJutJ

18 CONTENTS

5.4.5. Series of (he form V < * " * J s i n k*\ 730 • " (* 2 ±o' - ' ) m UosfcrJ

5.4.6. Series of the form V „. | s i n '=*+ ' » x l ' 731 _£j * lcos(2*+ \)xj

5.4.7. Series of the form V — / s i " kx\ 734 Z_J *l \cos*.t/

5.4.8. Series of the form V „. / s i n •** + 0 ) \ 735 JCJ * (cos(fct + o)/

5.4.9. Series of the form V — / s i n k*\ 736 • ^ (k + a)s l « » * x /

5.4.10. Series of the form V r* fsin (*jr+o)l 7 3 8

^_l *l \cot{kx + a)f

5.4.11. Series of the form V ahqa"" ls]nkx\ 739

5.4.12. Series of the form V abrka ihlnkx\ 739

5.4.13. Series of the form V o , r * I s i n fcr V 740

5.4.14. Seriesof the form V """ <*"">*) 741

5.4.15. Seriesof the form V ak W sin kxII coskyt 743

5.4.16. Various series contuining trigonometric funetions 745

5.5. Series Involving the Logarithnuc and Inverse Trigonometric Functions 746

5.5.1. Series of the form V a, In fct 746 — k k

5.5.2. Series of the form V , arctg*. 749

5.6. Multiple Series 750

Chapter 6. PRODUCTS 752

6.1. Finite Products 752

6.1.1. The power and algebraic funetions 752 6.1.2. Trigonometric funetions 752

6.2. Infinite Products 753

6.2.1. The power and algebraic funetions 753

6.2.2. Products containing (\±x")m 755 6.2.3. The exponemia! function 756 6.2.4. Trigonometric funetions 757

Appendix I. Some Elementary Functions and their Properties 758

1.1. Trigonometrie funetions 758 1.2. Hyperbolic funetions 765 1.3. Inverse trigonometric funetions 767 1.4. Inverse hyperbolic funetions 770 1.5. Binomial coefficients 772 1.6. The Pochhammer symbol (a)k 772

CONTENTS 19

Appendix II. Some Special Functions and their Properties 773

II. I. The gamma i'unction T(z) 773 11.2. The i'unction i|)(z) 774 11.3. The function ß(z) 775 11.4. The Riemann zeta function £(;) 776 11.5. The Bemoulli polynomials ß„(.v) 776 11.6. The Bemoulli numbers B„ 777 11.7. The Euler polynomials £„(*) 777 11.8. The Euler numbers £„ 777

Appendix III. Table of the Functions V, and V* 778

Bibliography 7S3 Index of nolations for functions and constants 787 Index of niathcmatical Symbols 794 Index of series expansions of some functions 795 Index of integral representalions of some special functions in terms of elemenlary l'unctions . . . . 798