φ[x_] := 
      If[-1 ≤ x ≤ 1, 1, 0]

Table[
      Plot[(1/2) (∫_(x - t)^(x + t))φ[τ] \[DifferentialD]τ, {x, -20, 20}, PlotRange -> {2, 0}, 
        PlotStyle -> {RGBColor[1, 0, 0], Thickness[0.01]}], {t, 0, 10, 
        0.1}]