integrate sqrt(1-x^2)?

Question

it's not the answer I'm interested in, but the way you got it.

Answer 1

Let x = sin θ, dx = cos θ dθ.
∫√(1-x^2) dx
= ∫√(1-sin^2 θ) cos θ dθ
= ∫cos^2 θ dθ
= ∫(cos 2θ + 1)/2 dθ
= 1/2 (sin 2θ/2 + θ) + c
= 1/2 (sin θ cos θ + θ) + c
= 1/2 (x√(1-x^2) + arcsin x) + c.