\[ \int_0^{1/2}\arcsin x\, dx=\int_0^{1/2}1\cdot\arcsin x\, dx= \] \[ \Big[x \arcsin x\Big]_0^{1/2}-\int_0^{1/2}\dfrac{x}{\sqrt{1-x^2}}\, dx= \]\[\dfrac12\cdot\dfrac{\pi}{6}+\dfrac12\int_0^{1/2}\dfrac{-2x}{\sqrt{1-x^2}}\, dx=\]\[ \dfrac{\pi}{12}+\dfrac12\Big[2\sqrt{1-x^2}\Big]_0^{1/2}=\]\[ \dfrac{\pi}{12}+\dfrac12(\sqrt3 -2)=\dfrac{\pi}{12}+\dfrac{\sqrt3}{2} -1.\]