Making a function out of repeated hyperbola integrations?

*To*: mathgroup at smc.vnet.net*Subject*: [mg122061] Making a function out of repeated hyperbola integrations?*From*: Nathan McKenzie <kenzidelx at gmail.com>*Date*: Tue, 11 Oct 2011 04:24:30 -0400 (EDT)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com

I'm working with the following repeated integrals. Is there any way to automate what I'm doing here? I start with this: Integrate[ n/x - a, {x, a, n/a}] The result of that (after a bit of text editing) is a^2 - n Log[a] + n (-1 + Log[n/a]). That's the first result I want to work with. Then, I currently manually edit that result by swapping out n with (n/x), and have Integrate[ a^2 - (n/x) Log[a] + (n/x) (-1 + Log[(n/x)/a]), {x, a, n/ a}] And that resolves to -a^3 + n Log[a] + n Log[a]^2 + 1/2 n Log[n/a^2]^2 + n (a - (1 + Log[a]) Log[n/a]). Which is the second result I want to work with. The, I manually swap out n with (n/x) and integrate again on {x, a, n/a}, and repeat this process ad nauseum. It doesn't take long before the number of n's for me to edit becomes really unwieldy and error prone... and ideally I would like to do this many times in a row (say, up 30 or 40) and have the results around for random use in other contexts. What I would really like to be able to do is just have some sort of function where I can type F[n,a,s], where s is the number of times integration is performed, and then n and a (which will be actual numbers) will get evaluated. I feel like the step where I swap out n with (n/x) points at something problematic, though. Is there any way in Mathematica for me to construct such a function and get my results automatically?

**Follow-Ups**:**Re: Making a function out of repeated hyperbola integrations?***From:*Nathan McKenzie <kenzidelx@gmail.com>

**Re: Making a function out of repeated hyperbola integrations?***From:*Jacopo Bertolotti <jacopo.bertolotti@gmail.com>

**Re: Making a function out of repeated hyperbola integrations?***From:*Patrick Scheibe <pscheibe@trm.uni-leipzig.de>