<p>Evaluate: \ 3\int \frac{\sin^{-1}\sqrt{x}}{\sqrt{1-x}} \, dx</p>
Step-by-Step Solution
Key Concept: Use substitution u = √x to convert the integrand into a form where integration by parts becomes effective. The key is recognizing that sin⁻¹(√x) paired with its derivative creates a manageable integration by parts scenario.
<p><strong>Step 1:</strong> Let u = √x, so x = u², dx = 2u du</p><p><strong>Step 2:</strong> Substitute into the integral:<br/>3∫ (sin⁻¹(u))/(√(1-u²)) · 2u du = 6∫ (u·sin⁻¹(u))/√(1-u²) du</p><p><strong>Step 3:</strong> Apply integration by parts with:<br/>• p = sin⁻¹(u), dp = du/√(1-u²)<br/>• dq = u du/√(1-u²), q = -√(1-u²)</p><p><strong>Step 4:</strong> Using integration by parts formula ∫p dq = pq - ∫q dp:<br/>6[-u√(1-u²)·sin⁻¹(u) + ∫√(1-u²) · du/√(1-u²)] du<br/>= 6[-u√(1-u²)·sin⁻¹(u) + ∫du]<br/>= 6[-u√(1-u²)·sin⁻¹(u) + u] + C</p><p><strong>Step 5:</strong> Substitute back u = √x:<br/>= 6[-√x·√(1-x)·sin⁻¹(√x) + √x] + C<br/>= 6√x[1 - √(1-x)·sin⁻¹(√x)] + C</p><p><strong>Or equivalently:</strong> 6√x - 6√x·√(1-x)·sin⁻¹(√x) + C</p><p>∴ Answer: <strong>6√x - 6√(x(1-x))·sin⁻¹(√x) + C</strong></p>
Correct Answer: 6