c=1;
A[p_]=Sin[p]/(Pi p);
B[p_]=(1-Cos[p])/(Pi p);

u[t_]:=ListPlot[Table[{x,NIntegrate[5A[p]Cos[p x]Cos[p c t]+4B[p] Sin[p x]Cos[p c t],{p,0.001,500}]},{x,-3,4,0.11}],PlotRange->{-0.2,5}]
u[0]

u[t_]:=ListPlot[Table[{x,NIntegrate[A[p]Cos[p x]Cos[p c t]+B[p] Sin[p x]Cos[p c t],{p,0.001,500}]},{x,-3,4,0.11}],PlotRange->{-0.2,1.2}];

Print["u(x,0):"];u[0]
Print["u(x,0.1):"];u[0.1]
Print["u(x,0.2):"];u[0.2]
Print["u(x,0.4):"];u[0.4]
Print["u(x,0.7):"];u[0.7]
Print["u(x,1):"];u[1]
Print["u(x,2):"];u[2]
Print["u(x,3):"];u[3]

A2[p_]=3Sin[p]/(Pi p);
B2[p_]=(1-Cos[p])/(Pi p);
AA[p_]=Sin[p]/(Pi p);
BB[p_]=3(1-Cos[p])/(Pi p);
u2[t_]:=ListPlot[Table[{x,NIntegrate[A2[p]Cos[p x]Cos[p c t]+B2[p] Sin[p x]Cos[p c t]+
AA[p]/(p c)Cos[p x]Sin[p c t]+BB[p]/(p c) Sin[p x]Sin[p c t],{p,0.001,500}]},{x,-5,5,0.101}],PlotRange->{-0.2,2.2},AspectRatio->1/2];

Print["u(x,0):"];u2[0]
Print["u(x,0.1):"];u2[0.1]
Print["u(x,0.2):"];u2[0.2]
Print["u(x,0.4):"];u2[0.4]
Print["u(x,0.7):"];u2[0.7]
Print["u(x,1):"];u2[1]
Print["u(x,2):"];u2[2]
Print["u(x,3):"];u2[3]