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