Issues with integrating a function

Question

I have the following function which I try to integrate:

lmbda0 = 132.19377;
a0 = 1.318;
Re0 = 2.235;
xe0 = a0*Re0;
alpha0 = 2 lmbda0 - 2 n0 - 1;

n0 = 5;

Plot[(Exp[
    0.5*(Log[n0!*alpha0] - Log[Gamma[2 lmbda0 - n0]]) + (lmbda0 - 
        n0 - 0.5) Log[2 lmbda0 Exp[-(a0*R - xe0)]] - 
     0.5 (2 lmbda0 *Exp[-(a0*R - xe0)]) + 
     Log[LaguerreL[n0, alpha0, 2 lmbda0 Exp[-(a0*R - xe0)]]]])^2
 , {R, 2, 2.5}]

Integrate[(Exp[
    0.5*(Log[n0!*alpha0] - Log[Gamma[2 lmbda0 - n0]]) + (lmbda0 - 
        n0 - 0.5) Log[2 lmbda0 Exp[-(a0*R - xe0)]] - 
     0.5 (2 lmbda0 *Exp[-(a0*R - xe0)]) + 
     Log[LaguerreL[n0, alpha0, 2 lmbda0 Exp[-(a0*R - xe0)]]]])^2, {R, 
  2, 2.5}]

The plot looks like what I would expect with reasonable values, however when I try to integrate it, I get lots of errors and usually an output of the integral itself where R is still a variable (even if I defined it as the integration variable, so it should work). Can someone tell me what I am doing wrong? Thank you!

Answer 1

The expression that OP plots and integrates contains Exp of large numbers. One solution (in this case) is to organize things so that the expression is only evaluated when R is already numeric:

lmbda0=132.19377;
a0=1.318;
Re0=2.235;
xe0=a0*Re0;
alpha0=2 lmbda0-2 n0-1;

n0=5;

f[R_?NumericQ]:=(Exp[0.5*(Log[n0!*alpha0]-Log[Gamma[2 lmbda0-n0]])+(lmbda0-n0-0.5) Log[2 lmbda0 Exp[-(a0*R-xe0)]]-0.5 (2 lmbda0*Exp[-(a0*R-xe0)])+Log[LaguerreL[n0,alpha0,2 lmbda0 Exp[-(a0*R-xe0)]]]])^2;

NIntegrate[f[R],{R,2,2.5}]
(* 0.737361 *)

Plot[f[R],{R,2,2.5}]
(* like OP *)

Also note that I used NIntegrate.