Sato-Tate measure moments

Question

Let $\mu=\frac{1}{\pi}\sqrt{1-\frac{x^2}{4}}$ be the Sato-Tate measure on the interval $[-2,2]$. Show that $$\displaystyle\int_{-2}^2 x^{2p+1}d\mu=0$$ and $$\displaystyle\int_{-2}^2 x^{2p}d\mu=\displaystyle\frac{1}{p+1}{2p\choose p}=C_p$$ The first integral is trivial since the function under the integral is odd. For the second one is there a a nice way to see that the integral satisfies the same recurrence as the Catalan number? I would like to avoid calculation if possible. Or is the a short computation that proves it?