Answer
$$ - 12$$
Work Step by Step
$$\eqalign{
& \int_1^4 {\left( {\frac{1}{{\sqrt t }} - 3\sqrt t } \right)dt} \cr
& {\text{radical properties}} \cr
& = \int_1^4 {\left( {\frac{1}{{{t^{1/2}}}} - 3{t^{1/2}}} \right)dt} \cr
& {\text{negative exponent}} \cr
& = \int_1^4 {\left( {{t^{ - 1/2}} - 3{t^{1/2}}} \right)dt} \cr
& {\text{find the antiderivative by the power rule}} \cr
& = \left( {\frac{{{t^{1/2}}}}{{1/2}} - \frac{{3{t^{3/2}}}}{{3/2}}} \right)_1^4 \cr
& = \left( {2{t^{1/2}} - 2{t^{3/2}}} \right)_1^4 \cr
& {\text{part 1 of fundamental theorem of calculus}} \cr
& = \left( {2{{\left( 4 \right)}^{1/2}} - 2{{\left( 4 \right)}^{3/2}}} \right) - \left( {2{{\left( 1 \right)}^{1/2}} - 2{{\left( 1 \right)}^{3/2}}} \right) \cr
& {\text{simplify}} \cr
& = \left( {2\left( 2 \right) - 2\left( 8 \right)} \right) - \left( {2\left( 1 \right) - 2\left( 1 \right)} \right) \cr
& = - 12 \cr} $$
