Goldstein in "Classical Mechanics" (1ed) obtains
$$-\int \sum_j \left(\frac {\partial V}{\partial q_j} \delta q_j + \frac{\partial V}{\partial \dot q_j} \delta \dot q_j\right) dt=-\delta \int V dt$$
from
$$- \int \sum_j \delta q_j \left( \frac {\partial V}{\partial q_j}- \frac {d}{dt} \frac {\partial V}{\partial \dot q_j}\right) dt.$$
Could you explain me a little about the steps?