Question

`∫ sin^(−1)` xdx is equal to ______.

Options

• `x sin^(−1) x + sqrt(1 − x^2)` + c

• `x sin^(−1) x − sqrt(1 − x^2)` + c

• cos⁻¹ x + c

• `1/sqrt(1 − x^2)` + c

Answer 1

`∫ sin^(−1)` xdx is equal to `bbunderline(x  sin^(−1) x − sqrt(1 − x^2) + c)`.

Explanation:

Let sin⁻¹ x = θ → x = sin θ

⇒ dx = cosθ dθ

∴ I = [θcos θdθ = θ cos θdθ + `∫((dθ)/(dθ)∫ cosθdθ)dθ`

= θ sinθ + ∫sinθdθ = θ sinθ − cosθ + c

= x `sin^(−1) x − sqrt(1 − x^2) + c`.