Answer
$\frac{{1011}}{2}$
Work Step by Step
$$\eqalign{
&\text{Let }I= \int_1^4 {\left( {8{t^3} - 6{t^{ - 2}}} \right)} dt \cr
& {\text{From Table 1 }}\left( {{\text{page 410}}} \right){\text{ }}\int {{x^n}} dx = \frac{{{x^{n + 1}}}}{{n + 1}} + C,{\text{ then}} \cr
& I = \left[ {\frac{{8{t^{3 + 1}}}}{{3 + 1}} - 6\left( {\frac{{{t^{ - 2 + 1}}}}{{ - 2 + 1}}} \right)} \right]_1^4 \cr
& = \left[ {\frac{{8{t^4}}}{4} - 6\left( {\frac{{{t^{ - 1}}}}{{ - 1}}} \right)} \right]_1^4 \cr
& = \left[ {2{t^4} + \frac{6}{t}} \right]_1^4 \cr
& {\text{Using the fundamental theorem of calculus}}{\text{, part 2}} \cr
&I = \left[ {2{{\left( 4 \right)}^4} + \frac{6}{{\left( 4 \right)}}} \right] - \left[ {2{{\left( 1 \right)}^4} + \frac{6}{{\left( 1 \right)}}} \right] \cr
& {\text{Simplifying}} \cr
& I = \frac{{1027}}{2} - 8 \cr
& = \frac{{1011}}{2} \cr} $$
