Compute $ \int\sin(x^2)\, dx + \int \sqrt{\arcsin t}\, dt$ Compute the sum of two integrals
$$
\int_{\large\sqrt{\frac{\pi}{6}}}^{\large\sqrt{\frac{\pi}{3}}}\sin(x^2)\ \ dx + 
\int_{\large\frac{1}{2}}^{\large\frac{\sqrt{3}}{2}} \sqrt{\arcsin t}\ \ dt.
$$

Let $x=\sqrt{\arcsin t}$, then
$$
\int\limits_{\sqrt{\frac{\pi}{6}}}^{\sqrt{\frac{\pi}{3}}}\sin(x^2)\, dx 
= \int\limits_{\frac{1}{2}}^{\frac{\sqrt{3}}{2}} t\,d\sqrt{\arcsin t}
= t\sqrt{\arcsin t}\vert_{\frac{1}{2}}^{\frac{\sqrt{3}}{2}} 
- \int\limits_{\frac{1}{2}}^{\frac{\sqrt{3}}{2}} \sqrt{\arcsin t}\, dt.
$$
