Answer
$$s\left( t \right) = - \frac{3}{2}\sin 2t + \frac{5}{2}t + 10$$
Work Step by Step
$$\eqalign{
& a\left( t \right) = 3\sin 2t;\,\,\,v\left( 0 \right) = 1,\,\,\,\,s\left( 0 \right) = 10 \cr
& v'\left( t \right) = a\left( t \right),{\text{ then}} \cr
& v\left( t \right) = \int {a\left( t \right)dt} \cr
& v\left( t \right) = \int {3\sin 2tdt} \cr
& v\left( t \right) = - \frac{3}{2}\cos 2t + C \cr
& {\text{Use the initial condition }}v\left( 0 \right) = 1 \cr
& 1 = - \frac{3}{2}\cos 2\left( 0 \right) + C \cr
& C = \frac{5}{2} \cr
& {\text{Thus}}{\text{, }} \cr
& v\left( t \right) = - \frac{3}{2}\cos 2t + \frac{5}{2} \cr
& s'\left( t \right) = v\left( t \right),{\text{ then}} \cr
& s\left( t \right) = \int {v\left( t \right)dt} \cr
& s\left( t \right) = \int {\left( { - \frac{3}{2}\cos 2t + \frac{5}{2}} \right)dt} \cr
& s\left( t \right) = - \frac{3}{2}\sin 2t + \frac{5}{2}t + C \cr
& {\text{Use the initial condition }}s\left( 0 \right) = 10 \cr
& 10 = - \frac{3}{2}\sin 2\left( 0 \right) + \frac{5}{2}\left( 0 \right) + C \cr
& C = 10 \cr
& {\text{Thus}}{\text{, }} \cr
& s\left( t \right) = - \frac{3}{2}\sin 2t + \frac{5}{2}t + 10 \cr} $$
