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integral of x^n

  • To: mathgroup at smc.vnet.net
  • Subject: [mg130575] integral of x^n
  • From: Richard Fateman <fateman at cs.berkeley.edu>
  • Date: Tue, 23 Apr 2013 00:03:24 -0400 (EDT)
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On 4/22/2013 12:13 AM, Alex Krasnov wrote:
> In this case, the results are valid for a=1 in the sense of a limit, as
> Limit[u, a -> 1] and limit(u, a, 1) demonstrate. This is not always the
> case. Example:
>
> In:	f = Integrate[x^n, x]
> Out:	x^(1 + n)/(1 + n)
>
> In:	Limit[f, n -> -1, Direction -> 1]
> Out:	-Infinity
>
> In:	Limit[f, n -> -1, Direction -> -1]
> Out:	Infinity
>
> Alex
>
>

Yes, but an equally valid antiderivative for x^n  is


s = (x^(n+1)-1)/(n+1).

note

Limit[s,n->-1]  is Log[x].

This alternative formula was, I think, pointed out more than once
to Wolfram Inc. probably circa version 2.

There are other issues that come up when using antiderivatives +
the fundamental theorem of integral calculus. Some of these become
apparent by reading FTIC carefully.

RJF





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