Answer
$$s\left( t \right) = 72{e^{ - t/6}} + 13t - 72$$
Work Step by Step
$$\eqalign{
& a\left( t \right) = 2{e^{ - t/6}};\,\,\,v\left( 0 \right) = 1,\,\,\,\,s\left( 0 \right) = 0 \cr
& v'\left( t \right) = a\left( t \right),{\text{ then}} \cr
& v\left( t \right) = \int {a\left( t \right)dt} \cr
& v\left( t \right) = \int {2{e^{ - t/6}}dt} \cr
& v\left( t \right) = 2\left( { - 6} \right){e^{ - t/6}} + C \cr
& v\left( t \right) = - 12{e^{ - t/6}} + C \cr
& {\text{Use the initial condition }}v\left( 0 \right) = 1 \cr
& 1 = - 12{e^{ - 0/6}} + C \cr
& C = 13 \cr
& {\text{Thus}}{\text{, }} \cr
& v\left( t \right) = - 12{e^{ - t/6}} + 13 \cr
& s'\left( t \right) = v\left( t \right),{\text{ then}} \cr
& s\left( t \right) = \int {v\left( t \right)dt} \cr
& s\left( t \right) = \int {\left( { - 12{e^{ - t/6}} + 13} \right)dt} \cr
& s\left( t \right) = - 12\left( { - 6} \right){e^{ - t/6}} + 13t + C \cr
& s\left( t \right) = 72{e^{ - t/6}} + 13t + C \cr
& {\text{Use the initial condition }}s\left( 0 \right) = 0 \cr
& 0 = 72{e^{ - 0/6}} + 13\left( 0 \right) + C \cr
& C = - 72 \cr
& {\text{Thus}}{\text{, }} \cr
& s\left( t \right) = 72{e^{ - t/6}} + 13t - 72 \cr} $$
