Answer
$$F\left( p \right) = \frac{{25{p^2}}}{2} + 3p - 53$$
Work Step by Step
$$\eqalign{
& f\left( p \right) = 25p + 3;\,\,\,F\left( 2 \right) = 3 \cr
& \cr
& {\text{Write a formula }}F\left( p \right){\text{ for the antiderivative of }}f\left( p \right) \cr
& F\left( p \right) = \int {\left( {25p + 3} \right)} dp \cr
& {\text{integrate}} \cr
& F\left( p \right) = \frac{{25{p^2}}}{2} + 3p + C \cr
& \cr
& {\text{Use the condition }}F\left( 2 \right) = 3{\text{ to find }}C \cr
& 3 = \frac{{25{{\left( 2 \right)}^2}}}{2} + 3\left( 2 \right) + C \cr
& 3 = 50 + 6 + C \cr
& 3 - 56 = C \cr
& C = - 53 \cr
& \cr
& {\text{The specific antiderivative of }}f{\text{ is}} \cr
& F\left( p \right) = \frac{{25{p^2}}}{2} + 3p - 53 \cr} $$
