Answer
$$2{t^{3/2}}{\bf{i}} + 2\ln \left| t \right|{\bf{j}} + t{\bf{k}} + {\bf{C}}$$
Work Step by Step
$$\eqalign{
& \int {\left( {3\sqrt t {\bf{i}} + \frac{2}{t}{\bf{j}} + {\bf{k}}} \right)} dt \cr
& {\text{ By the Definition of Integration of Vector - Valued Functions}} \cr
& = \left[ {\int {3\sqrt t dt} } \right]{\bf{i}} + \left[ {\int {\frac{2}{t}dt} } \right]{\bf{j}} + \left[ {\int {dt} } \right]{\bf{k}} \cr
& = 3\left[ {\int {{t^{1/2}}dt} } \right]{\bf{i}} + 2\left[ {\int {\frac{1}{t}dt} } \right]{\bf{j}} + \left[ {\int {dt} } \right]{\bf{k}} \cr
& {\text{Integrating }} \cr
& = 3\left( {\frac{{{t^{3/2}}}}{{3/2}}} \right){\bf{i}} + 2\ln \left| t \right|{\bf{j}} + t{\bf{k}} + {\bf{C}},{\text{ where }}{\bf{C}}{\text{ is a constant vector}} \cr
& = 2{t^{3/2}}{\bf{i}} + 2\ln \left| t \right|{\bf{j}} + t{\bf{k}} + {\bf{C}} \cr} $$
