       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 = %[]

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