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## 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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