Hi Orest,

An alternative to Martijn's answer

(* Acceleration *)
a[t_] := 
 Piecewise[{{0, t < 0}, {12, 0 <= t < 1}, {25, 1 <= t < 3}, {11, 3 <= t < 8}, {0, t >= 8}}]

(* Velocity *)
v[t_] := Integrate[a[t], t];

(* Distance *)
s[t_] := Integrate[v[t], t];

Plot[Evaluate[{a[t], v[t], s[t]}], {t, 0, 10},
 PlotTheme -> "Detailed",
 PlotRange -> All,
 ImageSize -> 600]

The 117 True in the Piecewise means the value is 117 everywhere else in the domain t > 8.

When you definve v[t_]:=Integrate [f[t],t] and you try to plot it you are basically solving for each t

Plot[{
    Integrate[f[0],0] 
    Integrate[f[1],1] 
    Integrate[f[2],2]
    ...
}]

That won't work, so first calculate the integral and then define a function that replaces after integration.

vt = Integrate[a[t], t] 
v[ti_] := vt /. t -> ti
Plot[v[t], {t, 0, 10}]