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Topic: The Charlwood Fifty
Replies: 52   Last Post: Jun 24, 2013 10:24 PM

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Albert D. Rich

Posts: 219
From: Hawaii Island
Registered: 5/30/09
Re: The Charlwood Fifty
Posted: Jun 18, 2013 3:19 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

On Monday, June 17, 2013 6:11:36 AM UTC-10, clicl...@freenet.de wrote:

> > On 6/16/2013 11:30 AM, clicliclic@freenet.de wrote:
> > > Oops, [problem #5] should have read:
>
> > > INT(COS(x)^2/SQRT(COS(x)^4 + COS(x)^2 + 1), x) =
> > > = x/3 + 1/3*ATAN(SIN(x)*COS(x)*(COS(x)^2 + 1)
> > > /(COS(x)^2*SQRT(COS(x)^4 + COS(x)^2 + 1) + 1))

>
> Apart from the compactness of antiderivatives, as measured by leaf
> counting, continuity on the real axis and absence of complex
> intermediate results when evaluated on the real axis (which implies
> absence of the imaginary unit) are important in my view, and usually
> take precedence over compactness.
>
> Thus, my 45-leafed result is fully continuous along the real axis,
> whereas the shorter ATAN alternative:
>
> INT(COS(x)^2/SQRT(COS(x)^4 + COS(x)^2 + 1), x) =
> = - 1/3*ATAN(COT(x)*COS(x)^2/SQRT(COS(x)^4 + COS(x)^2 + 1))
>
> as well as Albert's 37-leafed ASIN version:
>
> INT(COS(x)^2/SQRT(COS(x)^4 + COS(x)^2 + 1), x) =
> = - ASIN(COS(x)^3)*SQRT(1 - COS(x)^6)*CSC(x)
> /(3*SQRT(1 + COS(x)^2 + COS(x)^4))
>
> jump at x = -pi, 0, pi, etc. This constitutes an unnecessary obstacle in
> definite integration - imagine some quantity integrated along the path
> of an orbiting spacecraft.


I agree continuity of antiderivatives trumps compactness. Your impressive, continuous result for problem #5 is in the revised Charlwood Fifty pdf file at

http://www.apmaths.uwo.ca/~arich/CharlwoodIntegrationProblems.pdf

Please let me know if other problems can be continuitized, to coin a phrase.

Albert


Date Subject Author
5/23/13
Read The Charlwood Fifty
Albert D. Rich
5/23/13
Read Re: The Charlwood Fifty
Nasser Abbasi
5/23/13
Read Re: The Charlwood Fifty
clicliclic@freenet.de
6/16/13
Read Re: The Charlwood Fifty
clicliclic@freenet.de
6/16/13
Read Re: The Charlwood Fifty
Nasser Abbasi
6/17/13
Read Re: The Charlwood Fifty
clicliclic@freenet.de
6/17/13
Read Re: The Charlwood Fifty
Nasser Abbasi
6/17/13
Read Re: The Charlwood Fifty
clicliclic@freenet.de
6/17/13
Read Re: The Charlwood Fifty
Nasser Abbasi
6/17/13
Read Re: The Charlwood Fifty
Waldek Hebisch
6/18/13
Read Re: The Charlwood Fifty
Albert D. Rich
6/24/13
Read Re: The Charlwood Fifty
clicliclic@freenet.de
6/24/13
Read Re: The Charlwood Fifty
Albert D. Rich
5/23/13
Read Re: The Charlwood Fifty
Albert D. Rich
5/23/13
Read Re: The Charlwood Fifty
Waldek Hebisch
5/24/13
Read Re: The Charlwood Fifty
Andreas Dieckmann
5/25/13
Read Re: The Charlwood Fifty
clicliclic@freenet.de
6/1/13
Read Re: The Charlwood Fifty
clicliclic@freenet.de
6/3/13
Read Re: The Charlwood Fifty
Albert D. Rich
6/6/13
Read Re: The Charlwood Fifty
clicliclic@freenet.de
6/6/13
Read Re: The Charlwood Fifty
Nasser Abbasi
6/6/13
Read Re: The Charlwood Fifty
clicliclic@freenet.de
6/8/13
Read Re: The Charlwood Fifty
clicliclic@freenet.de
6/8/13
Read Re: The Charlwood Fifty
Albert D. Rich
6/8/13
Read Re: The Charlwood Fifty
clicliclic@freenet.de
6/8/13
Read Re: The Charlwood Fifty
clicliclic@freenet.de
6/9/13
Read Re: The Charlwood Fifty
Albert D. Rich
6/9/13
Read Re: The Charlwood Fifty
Nasser Abbasi
6/9/13
Read Re: The Charlwood Fifty
Albert D. Rich
6/9/13
Read Re: The Charlwood Fifty
Nasser Abbasi
6/10/13
Read Re: The Charlwood Fifty
clicliclic@freenet.de
6/10/13
Read Re: The Charlwood Fifty
Nasser Abbasi
6/10/13
Read Re: The Charlwood Fifty
Nasser Abbasi
6/12/13
Read Re: The Charlwood Fifty
Nasser Abbasi
6/13/13
Read Re: The Charlwood Fifty
clicliclic@freenet.de
6/13/13
Read Re: The Charlwood Fifty
Nasser Abbasi
6/13/13
Read Re: The Charlwood Fifty
Waldek Hebisch
6/13/13
Read Re: The Charlwood Fifty
Nasser Abbasi
6/13/13
Read Re: The Charlwood Fifty
Waldek Hebisch
6/14/13
Read Re: The Charlwood Fifty
clicliclic@freenet.de
6/14/13
Read Re: The Charlwood Fifty
Waldek Hebisch
6/14/13
Read Re: The Charlwood Fifty / Sage and sympy
Richard Fateman
6/14/13
Read Re: The Charlwood Fifty
clicliclic@freenet.de
6/9/13
Read Re: The Charlwood Fifty and Macsyma
Richard Fateman
6/9/13
Read Re: The Charlwood Fifty and Macsyma
Albert D. Rich
6/9/13
Read Re: The Charlwood Fifty, another Macsyma result
Richard Fateman
6/9/13
Read Re: The Charlwood Fifty, another Macsyma result
Albert D. Rich
6/10/13
Read Re: The Charlwood Fifty, another Macsyma result
Richard Fateman
6/10/13
Read Re: The Charlwood Fifty, another Macsyma result
clicliclic@freenet.de
6/10/13
Read Re: The Charlwood Fifty, another Macsyma result
Waldek Hebisch
6/11/13
Read Re: The Charlwood Fifty, another Macsyma result
Nasser Abbasi
6/11/13
Read Re: The Charlwood Fifty, another Macsyma result
clicliclic@freenet.de
6/11/13
Read Re: The Charlwood Fifty, another Macsyma result
Nasser Abbasi

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