The velocity of the moving particle is given by,

\( \displaystyle{\vec{v}=2\hat{i}+5t\hat{j}+\frac{1}{t}\hat{k}} \)

Since the velocity is the time derivative of the position vector  \( \vec{r} \), then we can write,

\( \displaystyle{\frac{d\vec{r}}{dt}=\vec{v}} \)

\( or,\ \displaystyle{\frac{d\vec{r}}{dt}=2\hat{i}+5t\hat{j}+\frac{1}{t}\hat{k}} \)

\( or,\ \displaystyle{d\vec{r}=\left[2\hat{i}+5t\hat{j}+\frac{1}{t}\hat{k}\right]dt} \)

\( or,\ \displaystyle{\vec{r}=\int{\left[2\hat{i}+5t\hat{j}+\frac{1}{t}\hat{k}\right]dt }} \)

\( or,\ \displaystyle{\vec{r}=2t\ \hat{i}+\frac{5}{2}t^2\ \hat{j}+ln{(t)}\ \hat{k}+c} \)

where, \( c \) is a constant.