Math documents

Integral of  xneax


The integral of xneax can be easily integrated by parts to produce a reduction formula that can also be expressed as a simple summation formula. More important is that the result can be used to evaluate many other difficult and cumbersome integrations that do not otherwise produce formulas with obvious general expressions.

 xn eax dx

 U = xn 
  
dV = eax dx
dU = n xn-1 
 
 V = 
eax x
a
 

xn eax
a
 - 
n
a
xn-1 eax dx
 
  For n > 1, continue expanding...
 
xn eax
a
 - 
n
a
(
xn-1 eax
a
 - 
n-1
a
xn-2 eax dx )
 
 
xn eax
a
 - 
n xn-1 eax
a2
 + 
n(n-1)
a2
xn-2 eax dx
 
xn eax
a
 - 
n xn-1 eax
a2
 + 
n(n-1)xn-2 eax
a3
 - 
n(n-1)(n-2)
a3
xn-3 eax dx
 
xn eax
a
 - 
n xn-1 eax
a2
 + 
n(n-1)xn-2 eax
a3
 - 
n(n-1)(n-2)xn-3 eax
a4
 + ... ± 
n! x eax
an
 ± 
n! eax
an+1
 
 
= eax
(
xn
a
 - 
n xn-1
a2
 + 
n(n-1)xn-2
a3
 - 
n(n-1)(n-2)xn-3
a4
 + ... ± 
n! x
an
 ± 
n!
an+1
)
 

There are n+1 terms.

The eax term can be factored out and the result can be expressed as

 xn eax dx = eax 
n
k=0
(-1)k 
n!
(n-k)!
xn-k
ak+1
 

which is similar to the Incomplete Gamma function. It is not my intention to go beyond this form of the series.



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