Answer
$$t{\bf{i}} + 3t{\bf{j}} + 2{t^2}{\bf{k}} + {\bf{C}}$$
Work Step by Step
$$\eqalign{
& \int {\left( {{\bf{i}} + 3{\bf{j}} + 4t{\bf{k}}} \right)} dt \cr
& {\text{ By the Definition of Integration of Vector - Valued Functions}} \cr
& = \left[ {\int {dt} } \right]{\bf{i}} + \left[ {\int {3dt} } \right]{\bf{j}} + \left[ {\int {4tdt} } \right]{\bf{k}} \cr
& {\text{Integrating }} \cr
& = t{\bf{i}} + 3t{\bf{j}} + 2{t^2}{\bf{k}} + {\bf{C}},{\text{ where }}{\bf{C}}{\text{ is a constant vector}} \cr} $$
