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Re: Simplifying Log to ArcCos Expressions

• To: mathgroup at smc.vnet.net
• Subject: [mg56901] Re: Simplifying Log to ArcCos Expressions
• From: Paul Abbott <paul at physics.uwa.edu.au>
• Date: Mon, 9 May 2005 01:46:50 -0400 (EDT)
• Organization: The University of Western Australia
• References: <d5kapv$299$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

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/

• Follow-Ups:
  • Re: Re: Simplifying Log to ArcCos Expressions
    • From: Andrzej Kozlowski <akoz@mimuw.edu.pl>