plotting – How to plot displacement-time graph when a piecewise function of velocity is given?

Given a piecewise function of v(t), how to plot its displacement x(t)?

ClearAll(v, x);
v(t_) := 2 t /; t < 10;
v(t_) := 20 /; 10 <= t < 25;
v(t_) := t - 5 /; 25 <= t < 35;
v(t_) := 30 /; 35 <= t < 50;
v(t_) := -7 t + 380 /; 50 <= t < 55;
v(t_) := -5 /; 55 <= t < 65;
x(t_) := Integrate(v(t), {t, 0, 10}) /; t <= 10;
x(t_) := Integrate(v(t), {t, 10, 25}) + x(10) /; 10 < t <= 25;
x(t_) := Integrate(v(t), {t, 25, 35}) + x(25) /; 25 < t <= 35;
x(t_) := Integrate(v(t), {t, 35, 50}) + x(35) /; 35 < t <= 50;
x(t_) := Integrate(v(t), {t, 50, 55}) + x(50) /; 50 < t <= 55;
x(t_) := Integrate(v(t), {t, 55, 65}) + x(55) /; 55 < t <= 65;
Plot({v(t), x(t)}, {t, 0, 65})

The above code produces a lot of errors:

Integrate::ilim: Invalid integration variable or limit(s) in
{0.00132786,0,10}.

NIntegrate::itraw: Raw object 0.0013278571428571428` cannot be used as
an iterator.

etc…