Question

integration of √sin^-1(√4x-1)

Answer 1

This requires Fresnel integral functions. It is not easy.

Let  4 x - 1 = sin² (t/2)
        4 dx = 2 sin (t/2) * cos (t/2)  dt / 2 =  Sin t  dt / 2

$I= \int {\sqrt{Sin^{-1}(\sqrt{4x-1})}} \, dx\\\\=\frac{1}{8\sqrt{2}}\int\ {\sqrt{t}*sin\ t} \, dt \\\\=\frac{1}{8\sqrt2}*[\sqrt{\frac{\pi}{2}}\ C(\sqrt{\frac{2}{\pi}}*\sqrt{t})-\sqrt{t}*Cos\ t]\\\\Here, C(x)=Fresnel\ Integral=\int\limits_0^x {cos(\frac{\pi y^2}{2})} \, dy$