The two formulations
$$\int_\Omega\bigl(\nabla\mathbf{v}\cdot\nabla\psi_i+\psi_i\nabla p \bigr) dV=0,$$
and 
$$\int_\Omega\bigl(\nabla\mathbf{v}\cdot\nabla\psi_i-p \nabla\psi_i\bigr) dV=0,$$
with $\psi_i$ a test function which vanishes on the boundary of $\Omega$, are both in use. (For the first formulation without partial integration of the pressure, see <A HREF="https://people.sc.fsu.edu/~jburkardt/classes/fem_2011/fem_ns3.pdf">Burkardt's lecture notes,</A> section 4.)    

The first formulation requires that both the velocity field and the pressure field must be $C^1(\Omega)$, in the second formulation the pressure can be $C^0(\Omega)$. There may be reasons where one requirement is favored over the other.