Answer
$$\sqrt 3 {\bf{i}} + \frac{1}{2}{\bf{j}} + \pi {\bf{k}}$$
Work Step by Step
$$\eqalign{
& \int_0^{\pi /3} {\left( {2\cos t{\bf{i}} + \sin t{\bf{j}} + 3{\bf{k}}} \right)} dt \cr
& {\text{Split the integrand }}\left( {{\text{See example 6, page 829}}} \right) \cr
& = \left( {\int_0^{\pi /3} {2\cos t} dt} \right){\bf{i}} + \left( {\int_0^{\pi /3} {\sin t} dt} \right){\bf{j}} + \left( {\int_0^{\pi /3} {3dt} } \right){\bf{k}} \cr
& = 2\left[ {\sin t} \right]_0^{\pi /3}{\bf{i}} - \left[ {\cos t} \right]_0^{\pi /3}{\bf{j}} + 3\left[ t \right]_0^{\pi /3}{\bf{k}} \cr
& {\text{Evaluating}} \cr
& = 2\left[ {\sin \left( {\frac{\pi }{3}} \right) - \sin \left( 0 \right)} \right]{\bf{i}} - \left[ {\cos \left( {\frac{\pi }{3}} \right) - \cos \left( 0 \right)} \right]{\bf{j}} + 3\left[ {\frac{\pi }{3} - 0} \right]{\bf{k}} \cr
& = 2\left( {\frac{{\sqrt 3 }}{2}} \right){\bf{i}} - \left( { - \frac{1}{2}} \right){\bf{j}} + \pi {\bf{k}} \cr
& = \sqrt 3 {\bf{i}} + \frac{1}{2}{\bf{j}} + \pi {\bf{k}} \cr} $$
