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MathGroup Archive 2005

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Re: Re: Simplifying Log to ArcCos Expressions

  • To: mathgroup at smc.vnet.net
  • Subject: [mg56912] Re: [mg56901] Re: Simplifying Log to ArcCos Expressions
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Tue, 10 May 2005 03:42:13 -0400 (EDT)
  • References: <d5kapv$299$1@smc.vnet.net> <200505090546.BAA13909@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

On 9 May 2005, at 14:46, Paul Abbott wrote:

> In article <d5kapv$299$1 at smc.vnet.net>,
>  "David Park" <djmp at earthlink.net> wrote:
>
>> I want to integrate the following expression and get a simple answer.
>>
>> expr1 = (k/r^2)*(1/Sqrt[1 - k^2/r^2])
>>
>> The answer is actually quite simple: ArcCos[k/r] + constant. But what 
>> a lot
>> of work for me to get it! Perhaps someone can show a simpler path. 
>> (I'm
>> working with Version 5.0.1.)
>
> Why not do what you would do by hand? That is, change variables:
>
> [1] k is a constant.
>
>   Dt[k] ^= 0;
>
> [2] Change variables r -> k/Cos[t]:
>
>   (k/r^2) (1/Sqrt[1 - k^2/r^2]) Dt[r] /. r -> k/Cos[t]
>
> and simplify.
>
>   Simplify[%, 0 < t < Pi/2]
>
> [3] Solve for t as a function of k and r.
>
>   Solve[r Cos[t] == k, t]
>
> Cheers,
> Paul
>
> -- 
> Paul Abbott                                      Phone: +61 8 6488 2734
> School of Physics, M013                            Fax: +61 8 6488 1014
> The University of Western Australia         (CRICOS Provider No 00126G)
> AUSTRALIA                               http://physics.uwa.edu.au/~paul
>         http://InternationalMathematicaSymposium.org/IMS2005/
>
>
I think the easiest way to do it using Mathematica and helping it "by 
hand" is to use the simple substitution

r -> k* s, where we can assume that s is positive. So we need to 
compute:


FullSimplify[k*Integrate[expr1 /. r -> k*s, s], s > 0]

-ArcCot[Sqrt[s^2 - 1]]

the rest is easy to do by hand using the well known formula

Cot[x]^2 ==Csc[x]^2-1

but it seems very hard to make Mathematica do this .


Andrzej Kozlowski




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