I would like to prove the following limit $$\lim\_{n\rightarrow +\infty}\int\_0^1(\cos\sqrt x)^n\,dx=0.$$

I know that I can prove it simply using the dominated convergence theorem, but I want to see if there are other ways, involving less sophisticated tools (theorems from basic calculus).

Can someone help me? Any hint would be highly appreciated! Thanks!