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									                                      List of Formulas in
      500 Integrals of Elementary & Special Functions
                                                         ╬

                                          Francis J. O’Brien, Jr., Ph.D.
                                           Aquidneck Indian Council
                                                  Newport, RI

                                                                                      April 2, 2013
© Francis J. O’ Brien, Jr.
All rights reserved.

                                               PREFACE
        This book is a listing of derived formulas for about 500 derivatives, integrals, series, and
products. The intended audience is students and practitioners in the fields of applied
mathematics, pure and applied science, and engineering. The fundamental purpose of the book
is to present a modest listing of formulas with worked out solutions structured on the widely used
desk reference, Table of Integrals, Series, and Products (7th ed.), Gradshteyn & Ryzhik (GR).
In essence, the book contains previously published formulas in GR as well as candidates for the
next edition.
        “New” formulas are provided in substantive areas—exponential and logarithmic
functions and selected special functions with emphasis on the gamma and related functions. The
level of difficulty of the material ranges from easy to moderately difficult, and covers selected
topics in first year calculus through advanced calculus/real variable anlysis. The focus is on
understanding how to evaluate the unsolved formulas covered in the text. Throughout the
computer is used as an answer–checker rather than the primary evaluator1.
        The topics covered reflect the author’s own interests in applied theoretical fundamentals
and potential applications in education, science and engineering. The first section, Mathematical
and Graphical Summary of Selected Elementary and Special Functions, contains an abbreviated
tutorial on all the notations, definitions and properties of the functions used in this book. The
Mathematical Summary should be consulted for guidance in the solution of the formulas
presented. The first major section of results consists of Indefinite Integrals of selected
Elementary Functions (Sections 2.3 & 2.7 in GR). This is followed by Definite Integrals of
Elementary Functions (Sections 3.3–3.4 & 4.2.–4.4 in GR), and lastly, selected Special
Functions (Sections 8.2 & 8.3 in GR). The primary contribution presented in this volume is a
family of integrals of three–parameter algebraic–exponential functions expressed in terms of
gamma functions. (see papers at http://www.docstoc.com/profile/waabu)
        As mentioned, 500 Integrals is based on the architecture of Table of Integrals, Series,
and Products. The material is presented in the order of appearance in GR using their section

1
    Mathematica can solve about 80% of the integrals in GR.
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nomenclature and formula numbering system. This format is used instead of conventional
chapter and sub–chapter headings.
NOTE: Some formulas in GR (7th Edition) contain misprints. For the errata sheet see
http://www.mathtable.com/errata/gr7_errata.pdf




        a n Pochhammer symbol               Li 2 dilogarithm
        C or  Euler' s constant              lim limit
        G or K Catalan' s constant            ln & log logarithm
         epsilon                             n! factorial
         not an element of a set             Re real variable
        Ei exponential - integral function     error function
        erf error function                     mathematical constant 3.14159265...
        erfi error function                    product
        exp or e exponential function          summation
        i imaginary number                     Psi (Digamma) function
         and  gamma function                n
                                                binomial coefficient
                                               j
          gamma function                     
        int integer function                  ! factorial
        li logarithm - integral function      !! double factorials




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                                               References

Abramowitz, Milton and Irene Stegun. Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables. Washington, DC: U.S.Government Printing Office, 1964
(Reprinted Dover, New York, 1972).

Artin, Emil. The Gamma Function. New York, NY: Holt, Rinehart and Winston, 1964.

Carr, G.S. Formulas and Theorems in Pure Mathematics, 2nd ed. New York: Chelsea
Publishing Co., 1970.

Gradshteyn, I.S. and I.M. Ryzhik (7th Edition). Table of Integrals, Series, and Products. Alan
Jeffrey and Daniel Zwillinger, Editors. NY: Academic Press, 2007.
Errata: http://www.mathtable.com/errata/gr7_errata.pdf


O’Brien, Francis J. Jr. 500 Integrals of Elementary and Special Functions, 2008.
ISBN: 1–4392-1981–8. http://www.docstoc.com/profile/waabu

                m  x n
_______.   x    e         dx and related integrals, 2nd ed., Mar.16, 2013.
http://www.docstoc.com/profile/waabu


_______. A family of algebraic-exponential integrals (corrected copy). Mar.g 26, 2013,
http://www.docstoc.com/profile/waabu




                                                     ✜
The author would like to acknowledge with deep gratitude the late Professor Alan Jeffrey for
extensive feedback in the creation of the formulas and assistance in their publication in
Gradshteyn and Ryzhik (6th & 7th Editions), Table of Integrals, Series, and Products.




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                       INDEX OF FORMULAS
                                ₪
               500 Integrals of Elementary & Special Functions
This index lists the formulas which are solved in the text. The formulas are identified by the section
number, section title and page numbers where they appear in the text. Similar expressions are stated
in combined algebraic form. The Mathematical and Graphical Summary (page 1 and following of
full text) contains other formulas not listed below.

           FORMULA             SECTION                 SECTION TITLE (page number)
                               NUMBER


                 NOTATION AND FINITE SUMS
int x                             —                         NOTATIONS (22)
a n                               —
 n                                  —
 uk
k m
 n                                  —
 f (k )
k m
n 1                              0.111                    FINITE SUMS (25–26)
 a  kr 
k 0
 n                                0.112
 aq k 1
k 1
 n                               0.121.1
k
k 1




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                              INDEFINITE INTEGRALS
                                            THE EXPONENTIAL FUNCTION (29–34)
x       e  ax dx
     m               n
                                   2.31

  e  ax
             n
                                   2.31
 x m dx
 exp a dx
         x
                                2.312

  ax                              2.312
 x a dx
  ax                              2.312
 x b dx
  2x                              2.312
 x 2 dx
 a dx
    x2                            2.312

  dx                              2.312
a   x   2




a
     xn
             dx                   2.312

  dx                              2.312
a   xn


x
     m
         a x dx
                 n
                                  2.312

     x                            2.312
d
dx 
     f (t )dt
     a
             c                    2.312
d
dx            f (t , a)dt
      ( x)

e                                           THE EXPONENTIAL COMBINED WITH
      x                          2.32
              ln p xdx
                                              RATIONAL FUNCTIONS OF x (35–60)
e
      x
              ln xdx
  e ax                             2.32
 x 4 dx
x
     m  ax n                      2.32
         e               dx
uv   vdu                         2.32




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              c                 2.32
d
dx             f (t )dt
       x 
       1!                   2.32
  k  1!
 ln xdx                        2.32
        m



 x ln xdx                      2.32
      n           m



 x ln xdx                      2.32
      n


erfi x                        2.32

e
      ax 2
              dx                2.32

                         dx
e
       ax 2 2 bx  c           2.32

a ,  a, x , a, x       2.32
  ln x n                        2.32
 x m dx
erf  x                        2.32

x
      m  x          n
                                2.32
          e               dx

x
      a 1  x                  2.32
              e dx
e
      x 2
              dx                2.32
                                2.32
    e  ax
                  n

      xm
                      dx
          n
  e ax                          2.32
 x dx
Eiax n                        2.32
    n
z  k  1!                    2.32
       z!
              2                 2.32
    e ax
     x2
                   dx

 e  x
              n
                                2.32
 x m dx
 n, x 
 e  x
              n
                                2.32
 x dx

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   
Ei  x n                                       2.32

      n
     x 2                                       2.32
  e
 x 2 dx
 x e dx                                         2.32
    n ax


  e ax                                           2.32
 x n dx
 x e dx
    n  x                                       2.32

 xe dx
          x                                    2.32

 x e dx
     
     2  x                                       2.32

 x e dx
     
   3  x                                         2.32

  e  x dx                                      2.32
 x
  e  x dx                                      2.32
 xn
  e  x dx                                      2.32
 x2
  e  x dx                                      2.32
 x3
  e  x dx                                      2.32
 x4
 x dx                                           2.32
    n


   1                                             2.32
x   n
          dx
   1                           1                 2.32
 x dx
     2
                   &        x   3
                                     dx

 x dx                                           2.32
     x



 11
                       n
                e ax                  1        2.32
                            1
  na            0
                            ln 
                            t
                                            dt

 d x
     x
dx
  e ax  1      e ax  1                         2.32
 eax 1  dx &  ax dx
                 e 1
 e ax  m                                        2.32
 eax n dx
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    e ax  m                    2.32
   e ax  n
                   dx

 e ax  m                       2.32
 eax n dx
 e ax  m                       2.32
 eax n dx
 e dx                          2.32
     x



 e dx
     x                         2.32

         dx                     2.32
 1 e        ax


   e ax                         2.32
 x  bdx
  x 1                          2.32
 e ax dx
     x                          2.32
 e x  1 dx
Li 2  z                       2.32
  lna  bx dx                 2.32
          x
  lnbx  a dx                 2.32
          x
  ln x dx                     2.32
 ax
Li 2 1                        2.32
Li 2  1

         dx                     2.32
      b
    a
      e mx
    dx                          2.32
 b
  a  mx
      e
     dx                         2.32
 a  bemx
  3  x
               2                2.32
x e      dx
  2  x                        2.32
x e     dx
Eix                           2.32


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      
           e t
                                     2.32
           t
                dt , x  0
     x
          e  t       t 
                          e
                                     2.32
 lim            dt       dt 
   0       t            t    
         x                    
 x0

l i x                              2.32
 x                                   2.32
    dt
  ln t , x  1
 0
       1 dt      x
                        dt 
                                     2.32
  lim                   
  0       ln t      ln t 
        0         1      
   x 1
      xe x                           2.32
  1  x 2 dx
      xe ax                          2.32
  1  ax 2 dx
       1                             2.32
       x2
    e 2 dx

 f (  x)  f ( x)                   2.32
   1 x2                           2.32
  e 2 dx
 
 f ( x)   f ( x)                  2.32
        1
          x2
                                     2.32
      xe 2 dx
 
  1
  
  2
 Eiax                              2.32
  x  1                           2.32

 f n  (x) for f ( x)  x n         2.32

             xn                      2.32
     lim          &
 x   e x
 L' Hôpital' s Rule
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 ln xdx                                   2.71                   LOGARITHMS (61–77)
      m



 lna  bx                               2.71
                                                     NOTE: ln n x  ln x 
                       m                                                  n
                           dx
   dx                                    2.72–2.73
 ln x
Eiln x                                 2.72–2.73

 x ln xdx                               2.72–2.73

 x ln xdx                               2.72–2.73
     n        m



 lnln x  dx                           2.72–2.73
                  n



 x lnln x  dx                         2.72–2.73
     n                    m



  ln ln x                              2.72–2.73
                  m

 x dx
     xn                                  2.72–2.73
 ln x m dx

         xn                              2.72–2.73
 ln x 2        dx

      xn                                 2.72–2.73
 ln x 3        dx

 a  bx  lnc  ln x                 2.72–2.73
                  m             n
                                    dx
  lnc  ln x                           2.72–2.73
                           n

 a  bx dx
 a  bx  lnc  kx  dx               2.72–2.73
           m           n


 lnc  kx                              2.72–2.73
                      n

 a  bx dx
         ln x n                          2.72–2.73
 a  bx         m
                          dx

 ln x n dx                               2.72–2.73
 a  bx
    ln x n dx                          2.72–2.73
   xm
  ln x                                   2.72–2.73
 x m dx


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    ln x 2 dx                2.72–2.73
     xm
    ln x 3                   2.72–2.73
     xm
               dx

x        lna  bx  dx
                      n
     m                         2.72–2.73
  ln a  bx dx               2.72–2.73
         x
       dx                      2.72–2.73
 lna  bx 
        x                      2.72–2.73
 lna  bx  dx
  x n dx                       2.72–2.73
 ln x
  ln m a  ln x              2.72–2.73
       xn
                  dx

  ln m a  ln x              2.72–2.73
        x
                  dx

x
     n
          ln m a  ln x dx   2.72–2.73

x
     n
          lna  ln x dx      2.72–2.73
    ln a  ln x              2.72–2.73
         x
                  dx

 x lna  ln x dx            2.72–2.73
     n


  ln a  ln x                2.72–2.73
       x
                dx
      dx                       2.72–2.73
 a  ln x n
     dx                        2.72–2.73
 a  ln x
      dx                       2.72–2.73
 a  ln x n
      dx                       2.72–2.73
x    ln x m
      n

     dx                        2.72–2.73
 x n ln x
       dx                      2.72–2.73
 a  ln x 2

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         dx
 x n ln x m
         dx                  2.72–2.73
 a  ln x    n


     dx                      2.72–2.73
 a  ln x
         xn                  2.72–2.73
 a  ln x    m
                    dx

     xn                      2.72–2.73
 a  ln x dx
      ln x                   2.72–2.73
 a  ln x 2 dx
     n
  e ax                       2.72–2.73
 x m dx into log function
 x e dx into log function
    m ax n                   2.72–2.73




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                               DEFINITE INTEGRALS
v                                 3.310        EXPONENTIAL FUNCTIONS (81–83)
 e dx
    x

u

  e  x  e a x                  3.310

0
        x
                 dx

Frullani Theorem                  3.310

Frullani Integral
                                 3.310
     f ( ax )  f (bx )
             x
                        dx
0

  e  ax  e  bx
              p            p
                                  3.310

0
         x
                  dx

     0
          xe x                    3.311

     
      1  e 2 x dx
          n
u
     ex       ln a                3.311


          x
                      dx

v                                 3.311
 a dx
          n
   x

u
1                                 3.311
 a dx
          n
   x

0
1
      1                           3.311
a
0
         xn
            dx

                                 3.311
      1
a
0
         xn
            dx

1                                 3.311
 a dx
    x

0
1                                 3.311
 a dx
          n
   x

0
u                                 3.321   EXPONENTIALS OF MORE COMPLICATED
 e dx
   x             2

                                                  ARGUMENTS (84–97)
0




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    2
        u                                                            3.321
         e dx
           x         2


        0
v                                                                    3.321
 e dx
   q x  2 2



u
                                                                    3.321
 e dx
   q x  2 2



u
                                                                    3.321
 e dx
   q x  2 2




u                                                                    3.321
 x e dx
   n q x         2 2



0
v
      x2                                                           3.322
  4  x dx
u
  exp 
     
            
            
                                                                    3.323
          
 exp  ax  bx dx
          2
                                           
0
                                                                    3.323
          
 exp  ax  bx dx
          2
                                           
u
                                                                     3.323
                                          
u

 exp  ax  bx dx
          2

0
v
        ax   2
                   kbx  c                                         3.323
e
   
                                     dx
u
v
        ax   2
                   2 bx  c                                        3.323
e
   
                                     dx
u

        ax   2
                   kbx  c                                         3.323
e
   
                                     dx
0

        ax   2
                   bx  c                                          3.323
e
   
                                 dx
0

        ax   2
                   bx  c                                          3.323
e
   
                                 dx
u
u
        ax   2
                   kbx  c                                         3.323
e
   
                                     dx
0

                                          ax               
 cx                                
                                                2
                                                     jbx
                                                                     3.323
         2
                   kgx e                                      dx





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
                                    ax                        
 cx                       
                                             2
                                                     bx
                                                                                     3.323
        2
                 gx e                                            dx
0

                                                     px                
 ax                               
                                                           2
                                                                kqx                 3.323
            2
                 bx  c e                                                 dx


                                 px                   
 ax                   
                                       2
                                            qx  c                                  3.323
        2
             kbx e                                       dx


                                                                                   3.323
 ax  kbx ce
                                              px  jqxr
                                                 2
        2
                                                                                dx




      ab                                                                          3.324
u

 exp  x 2 dx
0           
                                                                                     3.326
                                           
v

 x exp  x dx
   m        n

u
v
  exp   x n                                                                      3.326
 x m dx
u

                                           
                                                                                     3.326
 x exp  x dx
   m        n

0
v                                                                                    3.326
 x exp x  b dx
   m

u

                                                                  
                                                                                     3.326
 x exp  x dx
   m  n 1 n 1

0
                                                                                    3.326
 ( x  a )e
               ( x b)                        n
                         dx
0
                                                                                    3.326
 ( x  a )e
               ( x b)                        n
                         dx
u


                                                                                    3.326
 ( x  a )e
               ( x b)                        n
                         dx
0


                                                                                    3.326
 ( x  a )e
               ( x b)                        n
                         dx
u
u                                                                                    3.326
 ( x  a )e
               ( x b)                    n
                         dx
0




© Francis J. O’Brien, Jr., <> Aquidneck Indian Council<>All rights reserved.                 15
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                                                 3.327           EXPONENTIALS OF
                                
v

 exp  ae dx
          nx
                                                               EXPONENTIALS (99–102)
u
                                                3.327
      x  e  x 
 xe
0
                    dx

                                                3.327
      x  e  x 
 xe
0
                    dx

                                                 3.328
                        
v

 exp  e e dx
         x x

u
                                                 3.328
                            
v

 exp  e e dx
         x x

u

                                    
                                                 3.331
 x exp x  e dx
             x


                                                 3.331
                                            
v

 exp  e  x dx
          x

u
                                                 3.331
                                        
v

 exp  e  x dx
          x

u
                                                 3.331
                                        
v

 exp  e  x dx
          x

u


 exp e                                      3.331
                    x
                         x dx
0

                                                 3.331
                                            
v

 exp  e  x dx
          x

u
        
   x  1e  x dx
                                                 3.331
        1
        1                                        3.331
         x e dx
           1  x

        0
                                                3.331
   x  1e x dx
        0

                                                 3.331
     ,  
 d
d
                                                 3.331
    ,  
 d
d
v
  e  x                                         3.351   COMBINATIONS OF EXPONENTIALS AND
 x n1 dx
u
                                                            RATIONAL FUNCTIONS (103–105)



© Francis J. O’Brien, Jr., <> Aquidneck Indian Council<>All rights reserved.           16
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    
       ex                         3.351
    x dx
    u
u                                  3.351
 xe dx
     x

0
u                                  3.351
 x e dx
   2  x

0
u                                  3.351
 x e dx
   3  x

0
v                                  3.351
 x e dx
   n  x

u
v
        xe x                       3.353
 1  x 
u
               2
                   dx

      xe  x                       3.353
v

 1  x 
u
               2
                   dx

v
    e  qx                         3.361   COMBINATIONS OF EXPONENTIALS AND

u       x
             dx                              ALGEBRAIC FUNCTIONS (106–118)

    e  qx                         3.361

u       x
             dx

        e  x                     3.362
v


u       ax  b
               dx


        e  x                     3.362

u       x
               dx

        e  x                     3.362
u


0       x
               dx


      e  x                       3.362

u     xu
             dx

                                  3.363
 x x  u e dx
            x

u
v                                  3.363
 x a  bxe dx
            px

u
v                                  3.363
 xa  bx            e  px dx
                   m

u




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                                  3.363
          x

u    xu
                       e  x dx

                                  3.363
    x  px
 a  xe dx
0
v
      x                            3.363
 a  bx m e dx
               px

u

  ae  px
               2
                                   3.363
 a  x 2 dx
0

     x  u  qx                    3.363

u
      x
          e dx

         1                        3.371
     n
x
0
          2
              e  x dx

         1                        3.371
     n
x
0
          2
              e  x dx

    1                             3.371
         n
 x 2 e dx
        x

0
    n
          p                        3.371
x            e  x dx
          q

0
    n
          p                        3.371
x            e  x dx
          q

0
    p
       n                          3.371
x            e  x dx
     q

0
u                                  3.381   COMBINATIONS OF EXPONENTIALS AND
 x e dx
   m  x          n

                                              ARBITRARY POWERS (120–125)
0
                                  3.381
 x e dx
   m  x          n



u
                                  3.381
 x e dx
   m  x          n



0
u                                  3.381
 x e dx
    1  x

0

  ex                              3.381
 x dx
u




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v                                                  3.381
 x e dx
    1  x

u


   
        
            x 2 m exp   x 2 n dx               3.381

                                                  3.381
x e
                                   nb
  m  a  x

0
u                                                  3.381
x e
                                   n b
  m  a  x

0
                                                  3.381
x e
                                   nb
  m  a  x

u
                                                  3.382
 1  x  e dx
            x

0
    0
        1                     1  x               3.427   COMBINATIONS OF RATIONAL FUNCTIONS


    x  e
     
                           x
                                 1
                                   
                                     e dx                  OF POWERS AND EXPONENTIALS (126–128)

  e  vx  e  x
                   t                       t
                                                   3.434
 x  1 dx
0

  exp ax   exp bx dx
                                                  3.434

0
              x
                                                  3.461    COMBINATIONS OF EXPONENTIALS OF
 x e dx
   n 1  x                   2

                                                            MORE COMPLICATED ARGUMENTS AND
0                                                                   POWERS (129–160)
                                                  3.461
 x e dx
   2 n 1  x                     2



0
                                                  3.461
   2  n 1  x
 x e dx
                                       2



0

  e x
                   2 2
                                                   3.461
 x 2n dx
0
                                                  3.461
     e dx
        x            2




                              2                   3.461
      2 n  x
    x e       dx

v
         qx
               2   dx                              3.461
e
u                  x2
v                                                  3.462
 x e dx
                               n b
   m a  x

u



© Francis J. O’Brien, Jr., <> Aquidneck Indian Council<>All rights reserved.               19
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                                   3.462
 e dx
          na
    x

0
u                                   3.462
 e dx
          na
    x

0
                                   3.462
e
      x n  a
                  dx
u
                                   3.462
 x  b e
                        nb
            x
                               dx
0
v                                   3.462
 x  b e
                        n b
             x
                               dx
u
           n b
  e  x                            3.462
v

 x ma dx
u
          n b
  e  x                            3.462
 x ma dx
0
          n b
  e  x                            3.462
 x ma dx
u
           n b
  e  x                            3.462
u

 x ma dx
0
v                                   3.462
 ax  b  e dx
           m  x

u
v                                   3.462
 b  ax             e  x dx
                   m

u
                                   3.462
 b  ax  e dx
           m  x

0
u                                   3.462
 b  ax  e dx
           m  x

0
                                   3.462
 b  ax  e dx
           m  x

u

    e  x                          3.462
v

 ax  b n dx
u

     e  x                         3.462
v

 b  ax n dx
u




© Francis J. O’Brien, Jr., <> Aquidneck Indian Council<>All rights reserved.   20
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
     e  x                                              3.462
 b  ax n dx
0

     e  x                                              3.462
u

 b  ax n dx
0

     e  x                                              3.462
 b  ax n dx
u
                     nq
            xa 
              
                                                         3.462
v           b 
    e
  
u
          xa   m p       dx
           b
                                                        3.462
 x  a e
              x a 
                        dx
0
                                                        3.462
 ax  b  e dx
           m  px

0
                                                        3.462
 ax  b  e dx
           m  px

u

                                                         3.462
u

 ax  b             e  px dx
                   m

0


    e  px                                               3.462
 ax  b n dx
0

    e  px                                               3.462
 ax  b n dx
u

    e  px                                               3.462
u

 ax  b n dx
0
                                                        3.462
 x  a e
               x a 
                         dx
0
                                             n q
                    m p           xa                3.462
  xa                               

 b 
0    
                            e       b 
                                                    dx
                                             n q
                     m p           xa                3.462
  xa
u                                     

 b 
0    
                            e       b 
                                                    dx
                                             n q
                    m p           xa                3.462
  xa                               

 b 
u    
                            e       b 
                                                    dx



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                                                    3.462
  xa       xa
                   j                             k

  b  exp   b  dx
0              


  e  x
           n
                                                     3.462
 x m dx
0

  e  x
           n
u                                                    3.462
 x m dx
0

  e  x
           n
                                                     3.462
 x m dx
u



 e                            dx
                                                    3.464
        x 2
                 e  x
                           2

                                 2
u
                                 x
u
    x 2  x 2  dx                               3.464
 e      e       2
0                x
v
                 b                                 3.471
 exp  x
u    
                   n 
                     
                       dx

u
      b                                            3.471
 exp  x n dx
0           
u
      b                                            3.471
 exp  x dx
0         
u
      b                                            3.471
 exp  x 2 dx
0           
        dx                                        3.471
u

 exp  x n  x n1
0           

                                         dx
v                                                    3.471
 exp   ax  b 
                                     n

u


                                         dx
u                                                    3.471
 exp   ax  b 
                                     n

0


                                         
u                                                    3.471
 exp   ax  b  dx
                                     1

0



 exp  ax  b                        dx
u
                                     2
                                                     3.471

0
u
       ax  b                                   3.471
 exp ax  b
0    
                                           dx
                                          


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                                  dx
v                                        3.471
 exp   b  ax 
                              n

u


                                  dx
u                                        3.471
 exp   b  ax 
                              n

0



 exp  b  ax                 dx
u
                              1
                                         3.471

0


                                  dx
u                                        3.471
 exp   b  ax 
                              2

0
u
                                      3.471
 exp b  ax dx
0             
u
         b                             3.471
 exp a  x  dx
0            

                           dx        3.471
u
                         1
 exp   x
            n p

0

      b                                3.471
u exp  n            
      x               dx
 xm
0


                                        3.471
     
v        xn
  e
 x m dx
u
                                        3.471
     
         n
  e x
 x m dx
0
                                        3.471
     
        xn
  e
 x m dx
u

             b                         3.471
u

x       exp  n dx
     m

0            x 

v
    a  x  1 exp   dx            3.471

u        x  1
                         
                         
                                    
                                   x

              
                                        3.473
            m 1 2 n 1dx
 exp  x x
                  n

0



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
                  
                                 3.473
              m 1 2 n 1
 exp  x x
          n
                            dx
0

                  
                                 3.473
             1 2 m n 1
 exp  x x
          n
                           dx

               dx
0

    cos ax n                   3.781    TRIGONOMETRIC FUNCTIONS OF MORE
               
 sin ax n
  xm 
              
                
                                          COMPLICATED ARGUMENTS COMBINED
0                                            WITH POWERS (new formula)

v  dx                            4.211       LOGARITHMIC FUNCTIONS (161–178)

u  ln x
                                4.211
   dx
 ln x
e
v                                4.211
 ln xdx
u
v
   dx                            4.211
 x ln x
u
v
  x n dx                         4.211
 ln x
u
u
  x n dx                         4.211
 ln x
0
v
    xdx                          4.211
 ln x
u
v                                4.211
 x ln xdx
u
e                                4.211
 ln x dx
1
v                                4.211
 lnln x dx
u
v
    ln x  p dx                 4.211

u
       x
e
  ln xdx                         4.211
 x
1
e
    ln x  p dx                 4.211

1
       x

© Francis J. O’Brien, Jr., <> Aquidneck Indian Council<>All rights reserved.   24
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1
  ln xdx                                 4.212
 a  ln x
0
1
  ln xdx                                 4.212
 a  ln x
0
v
            dx                           4.212
 a  ln x 
u
                        2


v
      dx                                 4.212
 a  ln x n
u
v
    ln xdx                               4.212
 a  ln x n
u
v
    ln xdx                               4.212
 a  ln x n
u
v
    ln xdx                               4.212
 a  ln x 2
u
                  1                    4.215
   1
1

  ln x 
0       
                          dx

1
        dx                               4.215
      1
                 

     ln 
0

     x
                 n
                     1                  4.215
   1 2
1

  ln x  dx
0       
1
            dx                           4.215
                n
                     1
0
     1 2
     ln 
     x
                  1                    4.215
   1
v

  ln x 
u       
                          dx

                              1        4.215
      1
            1
    1
   
       ln                          dt
n 0  t 
z      1
                 dt                      4.215
    n         1
                            z 1

             ln 
        0

             t



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  ln ln x                                                 LOGARITHMS OF MORE
                                              4.229
 x 2 dx
1
                                                         COMPLICATED ARGUMENTS (178)

1
    
     ln x                                      4.241   COMBINATIONS OF LOGARITHMS AND
41 x x  1 dx                                         ALGEBRAIC FUNCTIONS (179–180)
                  n                            4.269   COMBINATIONS INVOLVING POWERS OF
   1  2 p 1
1

  ln x  x dx
0       
                                                        THE LOGARITHM AND OTHER POWERS
                                                                    (181–191)
        x p 1                                 4.269
1

                 n
                    dx
0
     1 2
     ln 
     x
1                                              4.272
 ln x  x dx
         n p

0
1                                              4.272
 ln x x dx
          p

0
                  1
                      n                       4.272
   1  2  1
1

  ln x  x dx
0       
                  n
                       1                       4.272
   1
1
                       2
  ln x 
                                1
                           x          dx
0       

    ln x  p  a  n1 dx                     4.272

1           x2

    ln x  p  a 1 dx                        4.272

1         x2

    ln x  p 1 dx                            4.272

1        x2
v
    ln x  p 1 dx                            4.272

u        x2
                                              4.272
      dx
 ln x  p x 2
1
                                              4.272
       dx
 ln x  p1 x 2
1
1                                              4.274
e             q
                      x
                                     n
                                          dx
    x 1  ln x 
                                      2
0




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
   1          1  dx              4.281       COMBINATIONS OF RATIONAL
  x  1  x ln x  x
1                 
                                           FUNCTIONS OF ln x AND POWERS (193–194)
                                  4.281
          dx
 x 2 ln p  ln x 
1

  x n  a 1                       4.281
u

 ln x dx
0

  x n 1                           4.281
u

 ln x dx
0

   1              1         dx   4.283
  x ln x  ln x1  ln x  x
1                          

1
  1        1                     4.283
  x  ln1  x  dx
0                
1                                  4.326     COMBINATIONS OF LOGARITHMIC
 lna  ln x x dx
                 1
                                            FUNCTIONS OF MORE COMPLICATED
0                                             ARGUMENTS AND POWERS (196)

                                  4.331       COMBINATIONS OF LOGARITHMS
 e ln x  dx
    x         n
                                                AND EXPONENTIALS (197–201)
0
v                                  4.331
 e ln xdx
    x

u
v                                  4.331
 e ln xdx
   x

u
v                                  4.337
e
      x
            lna  bx dx
u
                                  4.337
 e lna  bx dx
    x

0
                                  4.337
 e lna  bx dx
    x

u
                                  4.337
 e lna  bx dx
    x

0




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                        SPECIAL FUNCTIONS
 0, x                     8.212             THE EXPONENTIAL INTEGRAL
                                                   FUNCTION Ei(x) (203–209)
Ei x n                      8.212
Ei x                        8.212
   1                         8.212
Ei  n 
   x 
   1                         8.212
Ei  
   x
Ei x  y                     8.212
Ei  x  y                 8.212
   1                         8.212
  x y
Ei     
       
   x                         8.212
Ei  
   y
     
Eia x                        8.212
lix                          8.240         THE LOGARITHM INTEGRAL li(x) (211)
 0, ln x                  8.240
ln x
    e  t                       8.241       INTEGRAL REPRESENTATIONS (212–213)


  t dt
lia x                        8.241
 li xy                       8.241
erfc x                       8.250         THE PROBABILITY INTEGRAL, THE
                                          FRESNEL INTEGRALS   x , S ( x), C ( x) , THE
                                                 ERROR FUNCTION ERF(x),
                                            AND THE COMPLEMENTARY ERROR
                                               FUNCTION ERFC( x ) (214–220)
  xy                        8.252
  x                          8.252
 
  y
  
  y                          8.252
 
  2x 
 x y                        8.252
      
  a 

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x  y                       8.252
  x y                       8.252
        
   2 
 a  bx                     8.252
 a x                        8.252
 1 z                        8.313           THE GAMMA FUNCTION (EULER’S
     
  v                                            INTEGRAL OF THE SECOND
                                                      KIND): Γ(z) (221)
  z 1                       8.313
     
  v 
x  a                       8.331      FUNCTIONAL RELATIONS INVOLVING THE
                                               GAMMA FUNCTION (222–238)
x  n                       8.331
n  x                       8.331
n  x n  x               8.331
    1  x 
 x                         8.331
  x                          8.331
  
  2
  x                          8.331
1  
  2
 1 x                        8.331
      
  2 
m 1  m                  8.331
  x  1                      8.331
        
  2 
x  1                       8.331
 1                           8.331
  x 
 2       
x  x                    8.334
1  x 1  x              8.334
1  x  x                8.334
 x  x  1               8.334
            1                8.334
 x  x  
            2
       1                     8.334
 x  
       2


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        1 1                 8.334
  x     x 
        2 2         
        1         1         8.334
 x   x  
        2         2
2 x                          8.334
    x  x                    8.334
1    1
    2 2 
n  x  x  n              8.334
1  x  x  1              8.334
n  x  x  n              8.334
x 1  x                   8.334
x n  x                   8.334
sin x                          8.335
2 n 1                          8.335
   n
                                8.335
x  &
            1
                  definitions
          ( x)
n 1      k                   8.335
  z  
k 0      n
2 z , 3 z , nz         8.335
nz  b                       8.335
n 1
       k                      8.335
  n 
    
k 1
n 1
        k    k              8.335
  n 1  n 
    
k 1           
n  k                        8.339     PARTICULAR VALUES: FOR N A NATURAL
                                                   NUMBER (239–252)
n  1                        8.339
  n                           8.339
   
  2
 n                            8.339
  1 
 2 
     p                        8.339
 n  
     q
     1       3              8.339
 n   ,…,  n  
      2      4


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  p                                            8.339
       n
 q         
 1             3                             8.339
  n  ,…,   n 
 2             4 
  p
  
                                                 8.339
  q
          1 1                                8.339
 n    n 
          2 2   
 n  n                                        8.339
   
 2  2
n  x                                         8.339
 x  n 
  z ,0                                        8.350   THE INCOMPLETE GAMMA FUNCTION
                                                                DEFINITION (253–255)
a,                                           8.350
 a,0                                          8.350
 a,                                          8.350
 *  n, x                                     8.351
 n  k , x                                    8.352         SPECIAL CASES (256–260)
n  k , x                                     8.352
 n  k , x                                   8.352
0, x , 1, x 2, x , 3, x ,            8.352
4, x 
1, x , 2, x ,                            8.352
3, x , 4, x 
 1, x ,  2, x ,                          8.352
 3, x ,  4, x 
0, x ,  1, x ,                          8.352
 2, x ,  3, x ,
 4, x 
 1, x ,  2, x ,  3, x ,  4, x        8.352

 1, x ,  2, x ,  3, x ,  4, x    8.352
  , x                                        8.353   INTEGRAL REPRESENTATIONS (261–262)
  , x  y                                    8.353
  , xy 
   x                                           8.353
  , 
   y
          

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 , x                       8.353
   x                         8.353
  , 
   y
         
 , x  y                   8.353
a  k , x                   8.356          FUNCTIONAL RELATIONS (263–264)
 a  k , x                  8.356
da, x                       8.356
    da
d a, x                      8.356
    da
                               8.356
    erf  x 
 d
dx
    Eix  & d lix 
 d                             8.356
dx              dx
d a, x                      8.356
    dx
d x                         8.356
   dx
 d  1                        8.356
             
dx   x  
             
      x                        8.356
     exp xy dy
d
dx 
   0

  1 d   , u n 
                              8.356
  
n du n 
  1             
     d   , u n             8.356
n  du n 
d  , u n                  8.356
d n 
          
 d   , u n                 8.356
d  
  1        1               8.359                RELATIONSHIPS WITH
  , x k ,  , x k                          OTHER FUNCTIONS (265–272)
  2        2         
  1
  , ax  
           2                  8.359
  2        
   1                         8.359
  ,  x 
   2       


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   1                               8.359
  ,  x 2 
   2          
  1                                8.359
  ,x 
  2      
  1                                8.359
  , x 2 
  2        
  1  1 2           1 2         8.359
       2 , x    2 , x 
                        
erf                              8.359
   n                               8.359
  ,  x 
   2      
  n                                8.359
 , x 
  k 
  n                                8.359
  , x
  k 
      ln ln t                              THE PSI FUNCTION. EULER’S CONSTANT:
                                    8.367
               dt                          INTEGRAL REPRESENTATIONS (273–275)
    1    t2
                                    8.367
   1          1  dt
  t  1  t ln t  t
1                 
  
      1             1         dt   8.367
          
   1 
       t ln t ln t 1  ln t  t
                              
1
  1         1                      8.367
  t  ln1  t  dt
0                
    
                     
                                     8.367
  t exp t  e t dt
    
0
    et    et                       8.367
  t e  1 dt
 
         t
              
                                    8.367
     t  e  t 
 te dt  Ei(1)
0
                                    8.367
Ei1   te t e dt
                   t



           0




                                             END

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