∫1/sqrt(1-2x^2) dx?

Question

How is this done?

Answer 1

Substitute x = (1/√2) cos θ to get
1 - 2x^2 = 1 - 2(1/2) cos^2 θ = sin^2 θ
dx = (1/√2) (-sin θ dθ)

so the integral becomes
∫(1/sin θ) (1/√2) (-sin θ dθ)
= -(1/√2) θ + c
= -(1/√2) arccos (√2 x) + c.