Answer
$s(t)=\frac{1}{3\omega}(1-\cos^{3}(\omega t))$
Work Step by Step
$$s(t)=\int v(t)~dt$$
$$s(t)=\int \sin(\omega t)\cos^{2}(\omega t)~dt$$
$$s(t)=-\frac{\cos^{3}(\omega t)}{3\omega}+C$$
Find $C$:
$$s(0)=-\frac{\cos^{3}(\omega \cdot 0)}{3\omega}+C$$
$$s(0)=-\frac{1}{3\omega}+C$$
$$0=-\frac{1}{3\omega}+C$$
$$\frac{1}{3\omega}=C$$
so:
$$s(t)=-\frac{\cos^{3}(\omega t)}{3\omega}+\frac{1}{3\omega}$$
$$s(t)=\frac{1}{3\omega}(1-\cos^{3}(\omega t))$$
