Integral x^m exp(-Bx^n) and Related integrals. 2nd ed

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```					                                 m  x
 x e dx And Related Integrals
n

2nd ed.

╬

Francis J. O’Brien, Jr., Ph.D.
Aquidneck Indian Council
Newport, RI
January 3, 2013

Introduction

The following indefinite integral is published in Gradshteyn and Rhyzik, Table of
Integrals, Series, and Products, as Formula 2.33.10:

m  x n

 x e dx  

  , x n              1

t
 1  t
e dt ,
                       
n                      n x n
m 1
 
n

The purpose of this paper is to derive this useful integral and related forms. The
derivation highlights the tactic of “inspired guessing” and justification by differentiation using
Leibnitz’s Rule for moderately difficult integrals.

n
m  x
The first step is to decide how to convert the indefinite form,                           x e dx, to a definite
integral. We start by analogy with the simplest exponential integral,

x
e        dx.

Trial and error shows that the indefinite integral is equivalent to,


x
I  e         dx    e  t dt.
x

It can be tested by integrating each side and finding, I  e  x (ignoring the constant of
integration). The equivalence can also be verified by differentiating the definite form,


d              
  e  t dt ,
dx            
 x          

which by Leibnitz’s Rule for differentiating under the integral is of the form,

 a            
d                                  d  x  
   f (t )dt     f   x  dx 
dx                                          
 x         
where  ( x)  x, a  , f (t )  e t . This gives


d            
  et dt     e x dx   e x
dx  
           
            dx 
  x       

by the Fundamental Theorems of Integral Calculus. See the referenced YouTube lectures, and
Finney and Thomas (chapter 5), for review of single variable work.

m  x n
Derivation of    x e dx
n
The best approach is first to do a change of variable on  x m e  x dx .
1
 t n    m 1
Letting t  x n , dt  nx n 1dx, x    ,  
            ,
         n

m  x           1              1 t
 x e dx                    t
n
e dt.

n

Using the above simple exponential as a model,

I   e x dx    et dt ,
x
we form the hypothesis (an initial guess),


m  x n                 1  t                                                    m 1
t  1e  t dt ,  
1                         1
 x e dx        n 
 t e dt         n 
                               n
t  x n

NOTE: readers familiar with incomplete gamma functions recognize that the definite integral
can be expressed as:

           

1                 1  t              1

n             t        e dt  
n   
  , x n .
t  x n

To test the hypothesis, we differentiate the above integral by Leibnitz’s Rule. O’Brien,
500 Integrals, p. 21, gives the most common forms of Leibnitz’s Rule for
one and two variables. For this integral the following differentiation rule is appropriate (called
Rule 5, p. 21) for our parameters:


d ( x)
 f t ,  dt   f  ( x),  
d
.
dx                                                   dx
 ( x)

m 1 n
Applying this rule we find after simplifying and noting,   1                                      ,
n

                     
                            

d               1  t                                   n  1  x n d       
x n   x m e  x .
1                    1                                                                    n
    t e dt    
dx  n                                                 x       e
t  x n
 n                                             dx      
                     

This calculation confirms the hypothesis by differentiating the integral.

Summarizing the work flow,

n                 t  x n
x m e  x dx                                       1e t dt 
1
                                   
n 
t

1                    1e t dt   1   , x n 

n 
t                       
n  


t  x                 n
m 1
       .
n

Application

The most useful application of this derived integral in science and engineering is the
improper integral,


m  x n
x      e            dx
0
Formula 3.326.2 in Gradshteyn and Ryzhik.

We can solve this integral by applying the limits to the indefinite form of Formula
2.33.10 and using the properties of the gamma function for real variables.

               x      , n ,0  n ,

m  x n              1                                                       
x    e         dx  
n 
  , x n
x0
0
                   m 1    
Re    0, n  0,         0.
                    n      

The integral can also be evaluated by the change of variable form and definition of the complete
gamma function,

                     t  x n            
m  x n                     1             1  t      
x    e          dx      
n      t       e dt 
n 
,
0                                         0
                   m 1 
Re    0, n  0,        0
                    n     

To show one application for probability density functions (which is a common use of
Formula 3.326.2), we derive the Gaussian density by the method of this paper,

  1 x2
e 2 dx .

1             1
Since the Gaussian integral is an even function, and letting m  0,                                    , n  2,   , then,
2             2

1
  1 x2     1 x2                                           
 2   2 ,
e 2 dx  2 e 2 dx  2                                          1 
                            0                                1 2
2 
2
1
where     .
2

Other definite integrals can be derived by the indefinite form of 233.10. Some are
documented in O’Brien.

Related Integrals

Three related algebraic exponential integrals can be derived from the solution of
m  x n
 x e dx by substitution of parameters.
The following solution is an unpublished formula:

m ax n                11  ,ax n 
x          e           dx 
na 

 11                    1  t                       m 1

na            t        e dt  
n
 ax n

To derive this integral, set   a in Formula 2.33.10 and substitute,

m  x
 x e dx  
n              
  , x n            1

 1 t
 t e dt
  a

n                   n 
x n
                                      
1                  1 t            11                1 t

n a            t e dt              na             t      e dt
ax n                                  ax n
m 1

n

A related integral, Formula 2.33.19 in Gradshteyn and Rhyzik, is:

n                  z   z ,  x n 
e  x                               



 xm               dx  
n

z                 e t                             m 1

n                t z 1
dt                   z
n
x n

In Formula 2.33.10,

x
m  x n
e           dx  

  , x n             1

t
 1 t
e dt ,
                    
n                     n
x n
m 1

n

m 1
set m  m and define z              . The steps are omitted.
n

Lastly, the Gradshteyn and Rhyzik Formula 2.325.6 is given as:

 1z 1 a z  z,ax n 
n
e ax
     xm
dx 
n

 1z 1 a z                         1
e  t dt
m 1

n                    t   z 1
z
n
 ax n

To solve this form, in the derived integral, Formula 2.33.10

x
m  x n
e           dx  

  , x n             1

t
 1 t
e dt ,
                    
n                     n
x n
m 1

n

m 1
set m  m ,   a, define z                   , and substitute. The steps are omitted.
n

References

Finney, R.L. and G.B. Thomas, Jr. Calculus, 1990. Addison-Wesley Publishing Co.

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.

MIT 18.01. Single Variable Calculus, Fall 2007, Lectures 19 & 20.

O’Brien, Francis J. Jr. 500 Integrals of Elementary and Special Functions, 2008.
ISBN:1–4392-1981–8.
< http://www.docstoc.com/docs/23969109/500-Integrals-of-Elementary-and-Special-
Functions>.