Final answer:
To find the exact length of the curve y = x - x² sin⁻¹ x, use the arc length formula and make a substitution to simplify the integral.
Step-by-step explanation:
To find the exact length of the curve y = x - x² sin⁻¹ x, we can use the arc length formula. The arc length formula is given by L = ∫√(1 + (dy/dx)²) dx, where dy/dx represents the derivative of y with respect to x. In this case, the derivative of y with respect to x is 1 - 2x sin⁻¹ x - x² cos⁻¹ x.
To evaluate the integral, we need to make a substitution. Let u = sin⁻¹ x, then du = 1/√(1 - x²) dx. Substituting this into the expression for the derivative, we get 1 - 2sin u cos u - x² cos⁻¹ (sin u).
The integral becomes L = ∫√(1 + (1 - 2sin u cos u - x² cos⁻¹ (sin u))²) (1/√(1 - x²)) du. Simplifying this expression and evaluating the integral will give us the exact length of the curve.