I apologize if I misunderstood or wasn't clear.

In each of those the first line defines a function to do your integration and the second and third lines show example usage.

Asking Mathematica to evaluate this

i[tau_,phi_]:=Integrate[alpha E^(-t/tau)Cos[omega t+phi],{t,0,T}];
i[1,0]

then it will perform your integration with those two parameter values and should immediately display this

(alpha*(E^T - Cos[omega*T] + omega*Sin[omega*T]))/(E^T*(1 + omega^2))

Likewise asking Mathematica to evaluate

i[tau1_,tau2_,phi_]:=Integrate[(alpha1 E^(-t/tau1)+alpha2 E^(-t/tau2))Cos[omega t+phi],{t,0,T}];
i[1/2,1,Pi/2]

will perform your integration with those three parameter values and should immediately display

((alpha2*E^T*(-(E^T*omega) + omega*Cos[omega*T] + Sin[omega*T]))/(1 + omega^2) + 
 (alpha1*(-(E^(2*T)*omega) + omega*Cos[omega*T] + 2*Sin[omega*T]))/(4 + omega^2))/E^(2*T)

If I still don't have this correct then please let me know and we will try to resolve this.

Do these do what you want?

i[tau_,phi_]:=Integrate[alpha E^(-t/tau)Cos[omega t+phi],{t,0,T}];
i[1,0]
i[2,Pi/4]

and

i[tau1_,tau2_,phi_]:=Integrate[(alpha1 E^(-t/tau1)+alpha2 E^(-t/tau2))Cos[omega t+phi],{t,0,T}]
i[1,2,0]
i[1/2,1,Pi/2]