What is the indefinite integral of
√(1 - x^2) dx
3 answers
The indefinite integral of √(1 - x^2) dx is (1/2)(x√(1 - x^2) + sin^-1(x)) + C, where C is the constant of integration.
Can you show the steps ?
let x = sinθ, dx = cosθ dθ
∫√(1 - x^2) dx = ∫cosθ cosθ dθ
= ∫ (1+cos2θ)/2 dθ
= 1/2 (θ + 1/2 sin2θ) + C
now, sin2θ = 2sinθ cosθ = 2x√(1-x^2)
so we wind up with the given answer
∫√(1 - x^2) dx = ∫cosθ cosθ dθ
= ∫ (1+cos2θ)/2 dθ
= 1/2 (θ + 1/2 sin2θ) + C
now, sin2θ = 2sinθ cosθ = 2x√(1-x^2)
so we wind up with the given answer