Remove["Global`*"];
cr = 1.8*10^7/0.5*omega^2*Exp[-omega];
r55 = 2/Pi*Integrate[cr*Cos[omega*t], {omega, 0, 10}]

(*r55 = 1/(Pi (1.` + t^2)^3)
 2 (7.200000000000001`*^7 -
   2.1600000000000006`*^8 t^2 + (-199396.49151683348` -
      317073.10946119425` t^2 - 130751.79771595637` t^4) Cos[
     10 t] + (238622.0308316204` t + 388986.59820497024` t^3 +
      163439.7471449455` t^5) Sin[10 t]) *)


Remove["Global`*"];
omega = 1.2;

i1 = 25640;
ia = 4*10^6;
c2 = 1.4513*10^5;
t5 = 16000*Cos[omega*t];
tpto = 2.8*10^6*Theta'[t];

r55 = 1/(Pi (1.` + t^2)^3)
 2 (7.200000000000001`*^7 -
   2.1600000000000006`*^8 t^2 + (-199396.49151683348` -
      317073.10946119425` t^2 - 130751.79771595637` t^4) Cos[
     10 t] + (238622.0308316204` t + 388986.59820497024` t^3 +
      163439.7471449455` t^5) Sin[10 t]) /. {t -> t - Tau};

s = NDSolve[{(i1 + ia)*Theta''[t] +
 Integrate[r55*Theta'[Tau], {Tau, -Infinity, t}] +
 c2*Theta[t] == t5 + tpto, Theta'[0] == 1, Theta[0] ==
0}, Theta, {t, 0, 10}]