c=1;

A[p_]=1/Pi Integrate[(Cos[x]+1)Cos[p x],{x,-Pi,Pi}]
B[p_]=1/Pi Integrate[(Cos[x]+1)Sin[p x],{x,-Pi,Pi}]

u0[x_]=Integrate[A[p]Cos[p x],{p,0,3}]

Plot[{u0[x]},{x,-2Pi,2 Pi}]
Plot[{Cos[x]+1},{x,-2Pi,2 Pi}]

u[x_,t_]:=NIntegrate[A[p] Cos[p x]Cos[p c t],{p,0,3}]

plot0=Plot[u[x,0],{x,-10,10},PlotRange->{0,2},PlotPoints->5]
plot1=Plot[u[x,1.5],{x,-10,10},PlotRange->{0,2},PlotPoints->5]
plot3=Plot[u[x,3],{x,-10,10},PlotRange->{0,2},PlotPoints->5]
plot5=Plot[u[x,5],{x,-10,10},PlotRange->{0,2},PlotPoints->5]
plot7=Plot[u[x,7],{x,-10,10},PlotRange->{0,2},PlotPoints->5]
plot10=Plot[u[x,10],{x,-10,10},PlotRange->{0,2},PlotPoints->5]
Show[plot0,plot1,plot2,plot3,plot5,plot7,plot10]

Manipulate[Plot[u[x,t],{x,-10,10},PlotRange->{0,2},PlotPoints->7],{t,0,10,.5}]