I'm reading Grafakos' book on Fourier Analysis and at some point he says "There is an analogous calculation when $g$ is the characteristic function of the unit disk $B(0, 1)$ in $\mathbb{R}^2$. A simple computation gives$$(g * g)(x) = |B(0,1) \cap B(x,1)| = \int_{-\sqrt{1-\frac{1}{4} |x|^2}}^{+\sqrt{1-\frac{1}{4} |x|^2}}(2\sqrt{1-t^2} - |x|)dt \\= 2 \arcsin\left(\sqrt{1-\frac{1}{4}|x|^2}\right) - |x| \sqrt{1-\frac{1}{4}|x|^2}$$when $|x| \leq 2$ and $0$ otherwise". However, I wasn't able to reproduce this. Any hints?
Note. The operation $*$ above is the convolution.