Forcing function (non-homogeneous term)

f[t_]= Fo * Sin[π t] HeavisideTheta[t-3] HeavisideTheta[9-t];

p1=Plot[f[t]/.Fo->1,{t,0,20},PlotStyle->RGBColor[1,0,0]]

x[t_]= Integrate[Fo * Sin[π tp]* HeavisideTheta[t-tp]/(m ω1) *
Exp[-β(t-tp)]* Sin[ω1 (t-tp)],{tp,3,9}];

Response (heavy damping)

r={Fo->1,m->1,ω1->3,β->2};

p2=Plot[x[t]/.r,{t,0,20},PlotRange->All]

Show[p1,p2]

Response (light damping)

r={Fo->1,m->1,ω1->3,β->0.3};

p2=Plot[x[t]/.r,{t,0,20},PlotRange->All]

Show[p1,p2]