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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: 257 From: Hawaii Island Registered: 5/30/09
Re: The Charlwood Fifty
Posted: Jun 18, 2013 3:19 AM

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