Answer
$$F\left( z \right) = - \frac{1}{z} + \frac{{{z^2}}}{2} - \frac{1}{2}$$
Work Step by Step
$$\eqalign{
& f\left( z \right) = \frac{1}{{{z^2}}} + z;\,\,\,F\left( 2 \right) = 1 \cr
& {\text{Write a formula }}F\left( z \right){\text{ for the antiderivative of }}f\left( z \right) \cr
& F\left( z \right) = \int {\left( {\frac{1}{{{z^2}}} + z} \right)} dz \cr
& {\text{integrate}} \cr
& F\left( z \right) = - \frac{1}{z} + \frac{{{z^2}}}{2} + C \cr
& \cr
& {\text{Use the condition }}F\left( 2 \right) = 1{\text{ to find }}C \cr
& 1 = - \frac{1}{2} + \frac{{{2^2}}}{2} + C \cr
& 1 = - \frac{1}{2} + 2 + C \cr
& 1 = \frac{3}{2} + C \cr
& C = - \frac{1}{2} \cr
& \cr
& {\text{The specific antiderivative of }}f{\text{ is}} \cr
& F\left( z \right) = - \frac{1}{z} + \frac{{{z^2}}}{2} - \frac{1}{2} \cr} $$
