1 x-x^2=5/4-(x-1/2)^2
let
x-1/2=(√5/2)sinu
dx=(√5/2)cosudu
∫x/√(1 x-x^2)dx
=-(1/2)∫(1-2x)/√(1 x-x^2)dx (1/2)∫dx/√(1 x-x^2)
=-(1/2)√(1 x-x^2) (1/2)∫dx/√(1 x-x^2)
=-(1/2)√(1 x-x^2) (√5/5)∫du
=-(1/2)√(1 x-x^2) (√5/5)u C
=-(1/2)√(1 x-x^2) (√5/5)arcsin[(2x-1)/√5] C