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### Integration by Parts

```
Date: 12/8/95 at 5:12:43
From: "Juergen Meier"
Subject: Integration can be difficult

Suche Stammfunktion fuer :

x^n/e^x
```

```
Date: 5/30/96 at 11:16:51
From: Doctor Anthony
Subject: Re: Integration can be difficult

Using integration by parts, it is easy to get a reduction formula,
especially if the range of integration is from 0 to infinity.  I
shall assume these limits for the integration

Let I(n) = INT{x^n/e^x.dx} = INT{x^n.e^(-x).dx} (from 0 to infinity)

= x^n(-e^(-x)) + n.INT{x^(n-1).e^(-x).dx}   (from 0 to infinity)

=   0 + n.I(n-1)  and continuing in this manner

=  n(n-1).I(n-2)

Continue this process until, n(n-1)(n-2)...3.2.1.INT{e^(-x).dx}
(0 to inf.)

and this final integral is +1

So the value of the integral is n(n-1)(n-2)...3.2.1 = n!

If the limits are other than 0 to infinity, then there will be no
simple formula for the integral.  Provided n is not too large, it
can nevertheless be evaluated quite quickly.

-Doctor Anthony,  The Math Forum

```
Associated Topics:
High School Calculus

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