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

**References**:**Re: Simplifying Log to ArcCos Expressions***From:*Paul Abbott <paul@physics.uwa.edu.au>