from 0->1:
Int sqrt(1-x^2)

Solution)

I=∫sqrt(1-x²)dx = sqrt(1-x²)∫dx - ∫{(-2x)/2sqrt(1-x²)}∫dx ---->(INTEGRATION BY PARTS)        = x√(1-x²) - ∫-x²/√(1-x²)

Let I₁= ∫x²/√(1-x²) = ∫1-x²-1/√(1-x²) = ∫√(1-x²) - ∫1/√(1-x²) = ∫√(1-x²) - sin^-1x = I - sin^-1x

I = x√(1-x²) - I + sin^-1x

2I = x√(1-x²) + sin^-1x

I = x/2*√(1-x²) + 1/2*sin^-1x

Within limits 0 -> 1

I = [1/2*√(1-1) + 1/2sin^-11 - 1/2*√(1-0) - 1/2sin^-1(0) ] = 0 + pi/2 - 1/2 - 0 =  pi/2 - 1/2