Evaluate Tricky Multivariable Integral

Question

I'm trying to model an experiment for a paper which includes an integral that should be no problem for mathematica but it refuses to integrate it. The integral I need to evaluate is as follows

G[q_, t_] := (E^(2 I Sqrt[676000000 \[Pi]^2 -q^2] (23/1000000 + 
(1 - Cos[75000 t])/20000))q^3 ((8.46*^11 + 2.63*^11 I) 
Sqrt[1.88*^-13 - q^2] - Sqrt[(0.16 + 0.049 I) - q^2]))/(Sqrt[
676000000 \[Pi]^2 - q^2] ((8.46*^11 + 2.63*^11 I) Sqrt[1.88*^-13 
- q^2] + Sqrt[(0.15 + 0.049 I) - q^2]));

As you can see, it's a function of both q and t and I need to integrate it over q while keeping t constant as I have to integrate over t later on in the simulation. I have no idea why it won't integrate, please help!

Supplementary Information - this is me trying to solve the problem

The function G here is made up of two functions, rp, and another called Phi_{PDM}.

Phi_{PDM} integrated over q is no problem, it's also where the t dependence comes from, I've included here for completeness.

The problem lies with the function rp as I cannot get mathematica to integrate this function over q

As you can see, there is a singularity at small values of q which is probably whats causing the problem and I tried to mitigate this with assuming q is real and/or is above 10 but this still does not help. Here is the code for the equations:

Phi[q_, t_] := (q^3/(Sqrt[(2*\[Pi]*13000)^2 - q^2]))*Exp[I*2* 
(Sqrt[(2*\[Pi]*13000)^2 - q^2])*(23*^-6 + 50*^-6*(1 - 
Cos[75000*t]))];

rp[q_] := ((8.46*^11 + 2.63*^11 I) Sqrt[1.88*^-13 - q^2] - 
Sqrt[(0.16 + 0.049 I) - q^2])/((8.47*^11 + 2.63`*^11 I) Sqrt[
 1.88*^-13 - q^2] + Sqrt[(0.16 + 0.049 I) - q^2])

I'm not very good at mathematica and I'm certainly not the savviest when it comes to mathematics in general so any ideas regarding this, or indeed a method of solving G that I'm just not seeing due to me being a novice, would massively appreciate!

Answer 1

I solved the issue and thought it instructive to post the solution:

A contributor in the comments mentioned using NumericQ with NIntegrate, a function I had never previously come across. Turn out you can attach this to variable in an equation and it acts as a placeholder until such time that there is a number for that variable that then allows you to solve the problem numerically

rp[q_] := ((8.46*^11 + 2.63*^11 I) Sqrt[1.88*^-13 - q^2] - 
Sqrt[(0.16 + 0.049 I) - q^2])/((8.47*^11 + 2.63`*^11 I) Sqrt[
1.88*^-13 - q^2] + Sqrt[(0.16 + 0.049 I) - q^2])

\[Phi][\[Omega]_, q_, t_] := (q^3/(Sqrt[(2*\[Pi]*\[Omega])^2 - q^2]))*
Exp[I*2*(Sqrt[(2*\[Pi]*\[Omega])^2 - q^2])*z[t]];

G[\[Omega]_, t_?NumericQ] := 
NIntegrate[rp[\[Omega], q]*\[Phi][\[Omega], q, t], {q, 0, 2000000}]
array = ConstantArray[0, 50];

For[i = 1, i < 51, i++,
   array[[i]] = NIntegrate[Rationalize[FullSimplify[\[Alpha]tip[[i]]/(
   1 - \[Alpha]tip[[i]]*G[\[Nu][[i]], t])]]*Exp[I*5*75000*t], {q,0, 
   2000000}, {t, 0, 2 \[Pi]/75000}, AccuracyGoal -> 5];
   ]