Answer
$${\text{3}}t{\bf{i}} + 2{t^2}{\bf{j}} + {\bf{C}}$$
Work Step by Step
$$\eqalign{
& \int {\left( {3{\bf{i}} + 4t{\bf{j}}} \right)dt} \cr
& {\text{Integrating}} \cr
& {\bf{i}}\int 3 dt + {\bf{j}}\int {4t} dt \cr
& {\bf{i}}\left( {3t + {C_1}} \right) + {\bf{j}}\left( {2{t^2} + {C_1}} \right) \cr
& {\text{Simplifying, }}{\bf{C}} = {C_1}{\bf{i}} + {C_2}{\bf{j}}{\text{ is an arbitrary vector constant }} \cr
& {\text{3}}t{\bf{i}} + 2{t^2}{\bf{j}} + {\bf{C}} \cr} $$
