I read ∫cos^(1/2)xdx as ∫√cos(x)dx.
√cos(x) is a discontinuous function because cosine can be negative so the square root would be a complex number. The general indefinite integral cannot be defined because of discontinuity, and the integral is only defined for definite limits such as [-π/2,π/2]. y=√(1-4x²/π²) for −π/2≤x≤π/2 (semi-ellipse) is a close approximation (π²/4) to the definite integral ∫√cos(x)dx between the same limits.