Simplifying Log to ArcCos Expressions

*To*: mathgroup at smc.vnet.net*Subject*: [mg56866] Simplifying Log to ArcCos Expressions*From*: "David Park" <djmp at earthlink.net>*Date*: Sun, 8 May 2005 02:10:16 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

Dear MathGroup, 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.) expr2 = Integrate[expr1, r] -((Sqrt[k^2 - r^2]*Log[(2*(k + Sqrt[k^2 - r^2]))/ r])/(Sqrt[1 - k^2/r^2]*r)) Then I have to do all the following simplification steps... expr2[[{2, 3, 4}]] Numerator[%]/(Denominator[%] /. Sqrt[a_]*(b_) :> Sqrt[Distribute[a*b^2]]) % /. (a_)^(1/2)/(b_)^2^(-1) -> (a/b)^(1/2) Simplify[%, r >= k] Expand[%*FunctionExpand[expr2[[{1, 5}]]]] expr3 = %[[2]] expr3 MapAt[Distribute, %, {{2, 1}}] % /. Sqrt[a_]/(b_) :> Sqrt[Distribute[a/b^2]] % /. r -> k/z % /. Log[(z_) + Sqrt[(z_)^2 - 1]] -> I*ArcCos[z] % /. z -> k/r Thanks in advance for a more direct path. David Park djmp at earthlink.net http://home.earthlink.net/~djmp/

**Follow-Ups**:**Re: Simplifying Log to ArcCos Expressions***From:*DrBob <drbob@bigfoot.com>