(* Beispiel p.324: 1-Treppe *)

f[x_]:=NIntegrate[(Sin[p]/(Pi p) Cos[p x])+((1-Cos[p])/(Pi p) Sin[p x]),{p,0,1000}]

f[0];

ListPlot[Table[{x,NIntegrate[(Sin[p]/(Pi p) Cos[p x])+((1-Cos[p])/(Pi p) Sin[p x]),{p,0,500}]},{x,-2,2,0.05}],PlotRange->{-0.2,1.2}];

(* Beispiel mit 1-2-Treppe *)

g[x_]:=NIntegrate[3(Sin[p]/(Pi p) Cos[p x])+((1-Cos[p])/(Pi p) Sin[p x]),{p,0,1000}]

g[1.5];

ListPlot[Table[{x,NIntegrate[3(Sin[p]/(Pi p) Cos[p x])+((1-Cos[p])/(Pi p) Sin[p x]),{p,0,500}]},{x,-2,2,0.05}],PlotRange->{-0.2,2.2}];

Plot[NIntegrate[3(Sin[p]/(Pi p) Cos[p x])+((1-Cos[p])/(Pi p) Sin[p x]),{p,0,100}],{x,-2,2},PlotPoints->5,PlotRange->{-0.2,2.2}];

(* Beispiel f(x) = sgn(x)*Exp(-|x|) *)

Plot[Sign[x] Exp[-Abs[x]],{x,-2,2}];

ListPlot[Table[{x,NIntegrate[2p/(Pi (1+p^2)) Sin[p x],{p,0,500}]},{x,-2,2,0.07}],PlotRange->{-1,1}];