From: Robin Chapman <rjc@maths.ex.ac.uk>
Subject: Re: awkward integral
Date: Tue, 31 Aug 1999 13:02:08 GMT
Newsgroups: sci.math
Keywords: Use generating function to evaluate beta function definite integrals

In article <37C97FAE.ADFC9894@cableol.co.uk>,
  neenag@cableol.co.uk wrote:
>
>
> The question is:
>
> Find:
>
>  1
> Int x^k * (1-x)^(n-k) dx.
>  0

Use generating functions. Fix n and let I_k denote the integral.
Let F(t) = I_0 + (n choose 1) I_1 + (n choose 2) I_2 + ... + I_n
so that
F(t) = integral_0^1 (1+(t-1)x)^n dx
     = (t^{n+1} - 1)/[(n+1)(t-1)]
     = [1/(n+1)][1 + t + t^2 + ... + t^n].
Hence (n choose k) I_k = (n choose k)/(n+1) or
I_k = k!(n-k)!/(n+1)!.

--
Robin Chapman
 http://www.maths.ex.ac.uk/~rjc/rjc.html
  "They did not have proper palms at home in Exeter."
    Peter Carey, _Oscar and Lucinda_


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==============================================================================

From: Murray <mld@rosella.apana.org.au>
Subject: Re: awkward integral
Date: Wed, 01 Sep 1999 10:13:01 +1000
Newsgroups: sci.math

Herman Rubin wrote:
> 
> In article <7qgjo7$i1r$1@nnrp1.deja.com>,
> Robin Chapman  <rjc@maths.ex.ac.uk> wrote:
> >In article <37C97FAE.ADFC9894@cableol.co.uk>,
> >  neenag@cableol.co.uk wrote:
> 
> >> The question is:
> 
> >> Find:
> 
> >>  1
> >> Int x^k * (1-x)^(n-k) dx.
> >>  0
> 
> >Use generating functions. Fix n and let I_k denote the integral.

This is a Beta function, see eg Abramowitz & Stegun p258.

-- 
Murray
mld@junk.rosella.apana.org.au

Whether we get the right or wrong answer, we can now get it much faster
than a year ago.