Integrating the velocity vector

Question

If we integrate the velocity vector with respect to time, do we get the displacement vector or the position vector?

What I mean is that, do we integrate the velocity vector to get the displacement vector + constant, or can we say that position = integral of velocity vector + position at $(t=0)?$

Answer 1

If we integrate the velocity vector with respect to time, do we get the displacement vector or the position vector?

Integrating velocity over a time interval gives the particle's displacement across that period.

What I mean is that, do we integrate the velocity vector to get the displacement vector + constant,

The indefinite integral of velocity with respect to time gives the particle's position function up to an additive constant.

or can we say that position = integral of velocity vector + position at $(t=0)?$

Not exactly. The integral $\displaystyle\int_{t_1}^{t_2}\boldsymbol v\,\mathrm dt$ of the velocity vector over the interval $[t_1,t_2]$ is the displacement $\vec{P_1P_2},$ i.e., the position vector of ${P_2}$ minus the position vector of $P_1.$

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