Hi,

I meant that if you run

t1 = 1; t2 = 3;acceleration = Piecewise[{{t, 0 <= t < t1}, {0.5 t, t1 < t <= t1 + t2}}];
velocity = Integrate[acceleration, t]

you get the right result and can plot it:

Regarding the integration see final remark in:

https://pantherfile.uwm.edu/sorbello/www/classes/mathematica_badintegral.pdf

Cheers, Marco

Dear Seasong,

this is indeed not what you would expect at first. When you execute:

t1 =.; t2 =.;
acceleration = 
 Piecewise[{{t, 0 <= t < t1}, {0.5 t, t1 < t <= t1 + t2}}]
velocity = Integrate[acceleration, t]

you get

You see that the integral was evaluated for the three conditions individually. If you calculate

D[velocity, t]

you get the correct function of the acceleration, so that is ok. At the discontinuities there is obviously a problem with the derivative. Now if you run

t1 = 1; t2 = 3; Plot[NIntegrate[acceleration, {t, 0., tu}], {tu, 0, 4}]

you get

I guess that the problem is that when you execute the Integrate you have t1 and t2 not set to fixed values. If you run:

t1 = 1; t2 = 3;
acceleration = 
 Piecewise[{{t, 0 <= t < t1}, {0.5 t, t1 < t <= t1 + t2}}]
velocity = Integrate[acceleration, t]

you get

which gives the correct continuous plot.

Cheers, Marco