Answer
$$\frac{1}{2}{\bf{i}} + \frac{2}{3}{\bf{j}} + 2{\bf{k}}$$
Work Step by Step
$$\eqalign{
& \int_0^1 {\left( {t{\bf{i}} + \sqrt t {\bf{j}} + 4t{\bf{k}}} \right)} dt \cr
& {\text{Split the integrand }}\left( {{\text{See example 6, page 829}}} \right) \cr
& = \left( {\int_0^1 t dt} \right){\bf{i}} + \left( {\int_0^1 {\sqrt t } dt} \right){\bf{j}} + \left( {\int_0^1 {4t} dt} \right){\bf{k}} \cr
& = \left[ {\frac{1}{2}{t^2}} \right]_0^1{\bf{i}} + \left[ {\frac{2}{3}{t^{3/2}}} \right]_0^1{\bf{j}} + \left[ {2{t^2}} \right]_0^1{\bf{k}} \cr
& {\text{Evaluating}} \cr
& = \frac{1}{2}\left[ {{{\left( 1 \right)}^2} - {{\left( 0 \right)}^2}} \right]{\bf{i}} + \frac{2}{3}\left[ {{{\left( 1 \right)}^{3/2}} - {{\left( 0 \right)}^{3/2}}} \right]{\bf{j}} + 2\left[ {{{\left( 1 \right)}^2} - {{\left( 0 \right)}^2}} \right]{\bf{k}} \cr
& = \frac{1}{2}\left( 1 \right){\bf{i}} + \frac{2}{3}\left( 1 \right){\bf{j}} + 2\left( 1 \right){\bf{k}} \cr
& = \frac{1}{2}{\bf{i}} + \frac{2}{3}{\bf{j}} + 2{\bf{k}} \cr} $$
