Answer
$$2{x^4} + 2{x^{ - 1}} + 1$$
Work Step by Step
$$\eqalign{
& f\left( x \right) = 8{x^3} - 2{x^{ - 2}} \cr
& {\text{find an antiderivative of }}f\left( x \right),{\text{ use power rule}} \cr
& F\left( x \right) = 2{x^4} + 2{x^{ - 1}} + C \cr
& {\text{using the initial condition }}F\left( 1 \right) = 5 \cr
& 5 = 2{\left( 1 \right)^4} + 2{\left( 1 \right)^{ - 1}} + C \cr
& {\text{then}} \cr
& 5 = 2 + 2 + C \cr
& C = 1 \cr
& {\text{so}}{\text{,}} \cr
& = 2{x^4} + 2{x^{ - 1}} + 1 \cr} $$
