Particle dynamics, Newton laws

define:
\dot u=\frac{du}{dt},\ \ddot u=\frac{d^2u}{dt^2}
then:
F=ma=m\ddot x

v_{(t)}=\dot x=\int \ddot xdt =\\ =\frac{F_0}{m}\int\sin^2(\omega t) dt =\\=\frac{F_0}{2m}(t-\frac{\sin(2\omega t)}{2\omega})

x_{(t)}=\int vdt=\frac{F_0}{4m}t^2+\frac{F_0}{8m\omega^2}\cos(2\omega t)

where we assumed that
v_{(t=0)}=0,\ x_{(t=0)}=0