Answer
$$126{\bf{i}} + 510{\bf{j}} + 189{\bf{k}}$$
Work Step by Step
$$\eqalign{
& \int_1^4 {\left( {6{t^2}{\bf{i}} + 8{t^3}{\bf{j}} + 9{t^2}{\bf{k}}} \right)} dt \cr
& {\text{use }}\int_a^b {{\bf{r}}\left( t \right)} dt = \left( {\int_a^b {f\left( t \right)} dt} \right){\bf{i}} + \left( {\int_a^b {g\left( t \right)} dt} \right){\bf{j}} + \left( {\int_a^b {h\left( t \right)} dt} \right){\bf{k}}{\text{ }}\left( {{\text{see page 814}}} \right) \cr
& {\text{ then}} \cr
& = \left( {\int_1^4 {6{t^2}dt} } \right){\bf{i}} + \left( {\int_1^4 {8{t^3}dt} } \right){\bf{j}} + \left( {\int_1^4 {9{t^2}dt} } \right){\bf{k}} \cr
& {\text{evaluate integrals for each component}} \cr
& = \left( {\frac{{6{t^3}}}{3}} \right)_1^4{\bf{i}} + \left( {\frac{{8{t^4}}}{4}} \right)_1^4{\bf{j}} + \left( {\frac{{9{t^3}}}{3}} \right)_1^4{\bf{k}} \cr
& = \left( {2{t^3}} \right)_1^4{\bf{i}} + \left( {2{t^4}} \right)_1^4{\bf{j}} + \left( {3{t^3}} \right)_1^4{\bf{k}} \cr
& = \left( {2{{\left( 4 \right)}^3} - 2{{\left( 1 \right)}^3}} \right){\bf{i}} + \left( {2{{\left( 4 \right)}^4} - 2{{\left( 1 \right)}^4}} \right){\bf{j}} + \left( {3{{\left( 4 \right)}^3} - 3{{\left( 1 \right)}^3}} \right){\bf{k}} \cr
& {\text{simplifying}} \cr
& = 126{\bf{i}} + 510{\bf{j}} + 189{\bf{k}} \cr} $$
