r[x_] := Evaluate[q[x] /. NDSolve[{q'[t] == 0.0001 + (-1 + I*1 + q[t])*q[t],q[0] == 0}, q,
                                  {t, 0, 50}]]

fn[k_?NumericQ] := Exp[8*NIntegrate[r[s]*Exp[2*beta*1] + 8*r[s], {s, 0, k}]]

beta=1

NIntegrate[fn[k], {k, 0, 5}]

r[x_] := q[x] /.NDSolve[{q'[t] == 0.0001 + (-1 + I*1 + q[t])*q[t], q[0] == 0}, q, {t, 0, 50}][[1]]
beta = 1;
fn[k_?NumericQ] := Exp[8*NIntegrate[r[s]*Exp[2*beta*1] + 8*r[s], {s, 0, k}]]
NIntegrate[fn[k], {k, 0, 5}]
(*
5.07423 + 0.0503328 I
*)

eqns = {q'[t] == 10^-4 + (-1 + I*1 + q[t])*q[t], q[0] == 0};

sol = DSolve[eqns, q, t][[1]]

(* Solve::ifun: Inverse functions are being used by Solve, so some solutions may
   not be found; use Reduce for complete solution information.

{q -> Function[{t}, 1/100 ((50-50 I)+Sqrt[1+5000 I] Tan[1/100 (Sqrt[1+5000 I] t+
  100 ArcTan[(249950/25000001+(250050 I)/25000001) Sqrt[1+5000 I]])])]} *)

eqns /. sol // Simplify

(* {True, True} *)

r[x_] = q[x] /. sol;

beta = 1;

fn[k_?NumericQ] := Exp[8*NIntegrate[r[s]*Exp[2*beta*1] + 8*r[s], {s, 0, k}]]

NIntegrate[fn[k], {k, 0, 5}]

(* 5.07422 + 0.0503325 I *)