Question

Given \( \frac{d}{dx}F(x) = \frac{1}{\sqrt{2x - x^2}} \) and \( F(1) = 0 \), find \( F(x) \).

Hint:

For integrals involving square roots of quadratic expressions, use trigonometric substitutions to simplify.

Answer 1

Step 1: Express \( F(x) \) as an integral
The function \( F(x) \) is obtained by integrating the given derivative: \[ F(x) = \int \frac{1}{\sqrt{2x - x^2}} \, dx. \] Step 2: Simplify the expression inside the square root
Factorize \( 2x - x^2 \): \[ 2x - x^2 = x(2 - x). \] Thus: \[ F(x) = \int \frac{1}{\sqrt{x(2 - x)}} \, dx. \] Step 3: Substitute to simplify the integral
Let \( x = 1 - \sin^2 \theta \). Then: \[ 2 - x = 1 + \cos^2 \theta, \quad dx = -2\sin\theta\cos\theta \, d\theta. \] Substitute into the integral: \[ F(x) = \int \frac{1}{\sqrt{1 - \sin^2\theta}(1 + \cos^2\theta)} (-2\sin\theta\cos\theta) \, d\theta. \] Simplify and integrate: \[ F(x) = \sin^{-1}(x - 1) + C. \] Step 4: Use the initial condition to find \( C \)
Given \( F(1) = 0 \), substitute \( x = 1 \): \[ F(1) = \sin^{-1}(1 - 1) + C = 0 \implies C = 0. \] Step 5: Conclude the result
\[ F(x) = \sin^{-1}(x - 1). \]