from 0->1:Int sqrt(1-x^2) Solution)
I=∫sqrt(1-x^{2})dx = sqrt(1-x^{2})∫dx - ∫{(-2x)/2sqrt(1-x^{2})}∫dx ---->(INTEGRATION BY PARTS) = x√(1-x^{2}) - ∫-x^{2}/√(1-x^{2})
Let I_{1}= ∫x^{2}/√(1-x^{2}) = ∫1-x^{2}-1/√(1-x^{2}) = ∫√(1-x^{2}) - ∫1/√(1-x^{2}) = ∫√(1-x^{2}) - sin^{-1}x = I - sin^{-1}x
I = x√(1-x^{2}) - I + sin^{-1}x
2I = x√(1-x^{2}) + sin^{-1}x
I = x/2*√(1-x^{2}) + 1/2*sin^{-1}x
Within limits 0 -> 1
I = [1/2*√(1-1) + 1/2sin^{-1}1 - 1/2*√(1-0) - 1/2sin^-1(0) ] = 0 + pi/2 - 1/2 - 0 = pi/2 - 1/2