Answer
$$F\left( t \right) = \frac{{{t^3}}}{3} + {t^2} - 20$$
Work Step by Step
$$\eqalign{
& f\left( t \right) = {t^2} + 2t;\,\,\,F\left( {12} \right) = 700 \cr
& {\text{Write a formula }}F\left( t \right){\text{ for the antiderivative of }}f\left( t \right) \cr
& F\left( t \right) = \int {\left( {{t^2} + 2t} \right)} dt \cr
& {\text{integrate by using the power rule}} \cr
& F\left( t \right) = \frac{{{t^3}}}{3} + {t^2} + C \cr
& \cr
& {\text{Use the condition }}F\left( {12} \right) = 700{\text{ to find }}C \cr
& 700 = \frac{{{{\left( {12} \right)}^3}}}{3} + {\left( {12} \right)^2} + C \cr
& 700 = 576 + 144 + C \cr
& 700 = 720 + C \cr
& C = - 20 \cr
& \cr
& {\text{The specific antiderivative of }}f{\text{ is}} \cr
& F\left( t \right) = \frac{{{t^3}}}{3} + {t^2} - 20 \cr} $$