Definite Integration
Integration by Parts
Grade 12
Question:
<p>Let \(f: [-1, 0] \to \mathbb{R}\) be a function differentiable within the domain and that \(\displaystyle\int_{-1}^{0} (f(x))^2\, dx = 10\) and \(f(-1) = 2\). The value of the integral \(\displaystyle\int_{-1}^{0} x f'(x) f(x)\, dx\) is:</p>
<p>\(-1\)</p>
<p>\(-2\)</p>
<p>\(-3\)</p>
<p>\(-4\)</p>
Step-by-Step Solution
Key Concept: Use integration by parts with u = xf(x) and dv = f'(x)dx, recognizing that d/dx[f(x)]² = 2f(x)f'(x), which allows you to relate the unknown integral to the given integral of (f(x))².
<p><strong>Step 1:</strong> Apply integration by parts to ∫₋₁⁰ xf'(x)f(x)dx.</p><p>Let u = xf(x) and dv = f'(x)dx, so du = [f(x) + xf'(x)]dx and v = f(x).</p><p><strong>Step 2:</strong> By integration by parts:</p><p>∫₋₁⁰ xf'(x)f(x)dx = [xf²(x)]₋₁⁰ - ∫₋₁⁰ f(x)[f(x) + xf'(x)]dx</p><p><strong>Step 3:</strong> Evaluate the boundary term: [xf²(x)]₋₁⁰ = 0·f²(0) - (-1)·f²(-1) = 0 - (-1)·(2)² = 4</p><p><strong>Step 4:</strong> Expand the remaining integral:</p><p>∫₋₁⁰ xf'(x)f(x)dx = 4 - ∫₋₁⁰ [f(x)]² dx - ∫₋₁⁰ xf'(x)f(x)dx</p><p><strong>Step 5:</strong> Let I = ∫₋₁⁰ xf'(x)f(x)dx. Then:</p><p>I = 4 - 10 - I</p><p>2I = -6</p><p>I = -3</p><p>∴ Answer: D (which is -3)</p>
Correct Answer: D