A manufacturing process follows a learning curve where the time $T(n)$ to produce the $n$-th unit is $T(n) = T_1 \cdot n^{-b}$. If the first unit takes $100$ hours and the learning rate is $80\%$ (meaning $T(2n) = 0.8 T(n)$), what is the total time $S(N)$ to produce the first $N$ units, represented as an integral?