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

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 clicliclic@freenet.de Posts: 1,245 Registered: 4/26/08
Re: The Charlwood Fifty
Posted: Jun 16, 2013 12:30 PM
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clicliclic@freenet.de schrieb:
>
> here are my antiderivatives for problems #1 to #10, as promised. I
> think they are as compact, continuous, real, and elementary as one
> could wish them to be:
>
> [...]
>
> 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))
>

Oops, this 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))

Another simple elementarization of the elliptic problem #49 from
Charlwood's appendix is:

INT(ASIN(2*x*SQRT(1 - x^2)), x) =
= x*ASIN(2*x*SQRT(1 - x^2))
+ (1 - 2*x^2)/SQRT((1 - 2*x^2)^2)*(2*SQRT(1 - x^2) - SQRT(2))

Martin.

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