How to integrate a function with undefined variables?

Question

I'm trying to fit a function to data, but to make it go quicker I want to specify a starting point for Mathematica to start looking for the best fitting parameters. The equation I want to fit is:

SigmaBar[A_, Rs_][r_] :=  4 r NIntegrate[rho[A, Rs][u r] u^2, {u, 0, 1}] + 4 NIntegrate[rho[A, Rs][u] u/(u + Sqrt[u^2 - r^2]), {u, r, Infinity}]

where

 rho[A_, Rs_][r_] = (A 10^-3) r^(-1) (r + Rs)^(-2) Rs^(3);

I want to obtain starting values for two variables, $A$ and $R_s$. I have a list of values from my data for $\bar{\Sigma}$ and $r$, so I thought I could fill in two sets of those values (like $\bar{\Sigma}(r2)$, $r2$ and $\bar{\Sigma}(r4)$, $r4$), to solve for the $A$ and $R_s$. However, if I just use

Solve[{SigmaBar2 == SigmaBar[A, Rs][r2], SigmaBar4 == SigmaBar[A, Rs][r4]}, {A, Rs}]

I get an error saying that the integral in SigmaBar evaluated to non-numerical values (because A and Rs are not specified). Does anyone have an idea of how I can solve this?

Answer 1

Not a full answer, but a hint. You can get analytical solution for integrals with assumptions (fill in your special needs). This should be more easy to handle.

rhoA[A_, Rs_][r_] = (A 10^-3) r^(-1) (r + Rs)^(-2) Rs^(3);

Sig[A_, Rs_][r_] = 
    4 r Integrate[rhoA[A, Rs][u r] u^2, {u, 0, 1}, 
Assumptions -> 0 < A < 2 && 0 < Rs < 3 && 0 < r < 4] + 
    4 Integrate[
rhoA[A, Rs][u] u/(u + Sqrt[u^2 - r^2]), {u, r, Infinity}, 
Assumptions -> 0 < A < 2 && 0 < Rs < 3 && 0 < r < 4
 ]

(*   -((A Rs^3 (r + (r + Rs) Log[Rs/(r + Rs)]))/(250 r^2 (r + Rs))) + (
 A Rs^3 (r Sqrt[(r - Rs) (r + Rs)] + (r + 
   Rs) (Sqrt[r^2 - Rs^2] Log[r/(2 (r + Rs))] + 
   I Rs (Log[-((2 I r)/Sqrt[(r - Rs) (r + Rs)])] - 
      Log[2 - (2 I Rs)/Sqrt[r^2 - Rs^2]]))))/(
 250 r^2 Sqrt[r - Rs] (r + Rs)^(3/2))   *)