Answer
$$\frac{{{x^6}}}{6} + \frac{2}{x} + x - \frac{{19}}{6}$$
Work Step by Step
$$\eqalign{
& f\left( x \right) = {x^5} - 2{x^{ - 2}} + 1 \cr
& {\text{find an antiderivative of }}f\left( x \right),{\text{ use power rule}} \cr
& F\left( x \right) = \frac{{{x^6}}}{6} - 2\left( {\frac{{{x^{ - 1}}}}{{ - 1}}} \right) + x + C \cr
& F\left( x \right) = \frac{{{x^6}}}{6} + 2{x^{ - 1}} + x + C \cr
& {\text{using the initial condition }}F\left( 1 \right) = 0 \cr
& 0 = \frac{{{{\left( 1 \right)}^6}}}{6} + 2{\left( 1 \right)^{ - 1}} + 1 + C \cr
& {\text{then}} \cr
& C = - \frac{{19}}{6} \cr
& {\text{so}}{\text{,}} \cr
& = \frac{{{x^6}}}{6} + 2{x^{ - 1}} + x - \frac{{19}}{6} \cr
& = \frac{{{x^6}}}{6} + \frac{2}{x} + x - \frac{{19}}{6} \cr} $$
