Answer
$$2{e^u} + 3u + 6$$
Work Step by Step
$$\eqalign{
& f\left( u \right) = 2{e^u} + 3 \cr
& {\text{find an antiderivative of }}f\left( u \right) \cr
& F\left( u \right) = 2{e^u} + 3u + C \cr
& {\text{using the initial condition }}F\left( 0 \right) = 8 \cr
& 8 = 2{e^0} + 3\left( 0 \right) + C \cr
& 8 = 2\left( 1 \right) + 3\left( 0 \right) + C \cr
& 8 = 2 + C \cr
& 8 - 2 = C \cr
& C = 6 \cr
& {\text{then}} \cr
& = 2{e^u} + 3u + 6 \cr} $$
