Re: Simplifying Log to ArcCos Expressions

*To*: mathgroup at smc.vnet.net*Subject*: [mg56888] Re: [mg56866] Simplifying Log to ArcCos Expressions*From*: DrBob <drbob at bigfoot.com>*Date*: Mon, 9 May 2005 01:46:10 -0400 (EDT)*References*: <200505080610.CAA02252@smc.vnet.net>*Reply-to*: drbob at bigfoot.com*Sender*: owner-wri-mathgroup at wolfram.com

How about substituting k/r == Cos[t] BEFORE integrating? expr1 = (k/r^2)*(1/Sqrt[1 - k^2/r^2]) D[k/Cos@t, t]expr1 /. r -> k/Cos@t // Simplify // PowerExpand Integrate[%, t] + C % /. t -> ArcCos[k/r] k/(Sqrt[1 - k^2/r^2]*r^2) 1 C + t C + ArcCos[k/r] Sin[t] would work equally well. Bobby On Sun, 8 May 2005 02:10:16 -0400 (EDT), David Park <djmp at earthlink.net> wrote: > 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/ > > > > > -- DrBob at bigfoot.com

**References**:**Simplifying Log to ArcCos Expressions***From:*"David Park" <djmp@earthlink.net>