Question

a(t)=2t+3 v(0)=-4 [0,3] find the velocity and time t and the distance traveled during the given time interval

Answer 1

I would assume that a(t) represents the acceleration, if so then:
\[v(t)=\int\limits a(t)dt=\int\limits (2t+3)dt=t^2+3t+c\]
But v(0)=-4 implies c=-4. So the velocity is given by:
\[v(t)=t^2+3t-4.\]

Answer 2

Now to find the distance travelled during [0,3], integrate v(t) over the given interval. i.e calculate:
\[\int\limits_0^3(t^2+3t-4)dt\]