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