Chapter 8: Techniques of Integration - Section 8.6 - Integral Tables and Computer Algebra Systems - Exercises 8.6 - Page 481: 30

$2\sqrt x\cos^{-1}\sqrt x-2\sqrt{1-x}+C$

Let $I=\int\dfrac{\cos^{-1}\sqrt x}{\sqrt x}dx$ Suppose $u=\sqrt x \implies du=\dfrac{1}{2\sqrt x}dx $ or, $\dfrac{1}{\sqrt x}dx=2du$ So, $I=2 \space \int\cos^{-1}udu \\=2 (u\cos^{-1}u-(1/1)\sqrt{1-u^2}+C) \\=2\sqrt x\cos^{-1}\sqrt x-2\sqrt{1-x}+C$