Question

The value of `int (x sin^-1)/(sqrt(1 - x^2)) dx` is equal to:

Options

• `sqrt((1 - x^2)) sin^-1 x + C`

• x sin⁻¹ x + C

• `x - sqrt((1 - x^2)) sin^-1 x + C`

• `sqrt((sin^-1 x)) + C`

Answer 1

`bb(x - sqrt((1 - x^2)) sin^-1 x + C)`

Explanation:

Let I = `int (x sin^-1)/(sqrt(1 - x^2)) dx`

Take, sin⁻¹ x = t ⇒ x = sin t

`1/(sqrt(1 - x^2)) dx = dt ∴ cos t = sqrt(1 - x^2)`

I = ∫ sin t.t dt = t.(−cos t) − ∫ (−cos t)dt

= t cos t + sin t + C = `-(sqrt(1 - x^2)) sin^-1 x + x + C`