Answer
$$\frac{1}{3}{t^3}{\bf{i}} + \frac{5}{2}{t^2}{\bf{j}} + 2{t^4}{\bf{k}} + {\bf{C}}$$
Work Step by Step
$$\eqalign{
& \int {\left( {{t^2}{\bf{i}} + 5t{\bf{j}} + 8{t^3}{\bf{k}}} \right)} dt \cr
& {\text{ By the Definition of Integration of Vector - Valued Functions}} \cr
& = \left[ {\int {{t^2}dt} } \right]{\bf{i}} + \left[ {\int {5tdt} } \right]{\bf{j}} + \left[ {\int {8{t^3}dt} } \right]{\bf{k}} \cr
& {\text{Integrating }} \cr
& = \frac{1}{3}{t^3}{\bf{i}} + \frac{5}{2}{t^2}{\bf{j}} + 2{t^4}{\bf{k}} + {\bf{C}},{\text{ where }}{\bf{C}}{\text{ is a constant vector}} \cr} $$
