Evaluate: $\sum_{j=1}^{k}\cos(j^{1/k})$

Question

Where $\lfloor n \rfloor $ is floor function

How to show that if possible? $$\int_{0}^{1}\lfloor \arccos^k(x)\rfloor \mathrm dx=\sum_{j=1}^{k}\cos(j^{1/k})\tag1$$

and does $\sum_{j=1}^{k}\cos(j^{1/k})$ has a closed form?