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 on ∫x f'(x)f(x)dx by recognizing that f'(x)f(x) = ½d/dx[f(x)²], then apply the product rule strategically to isolate the unknown integral.
<p><strong>Step 1:</strong> Recognize that f'(x)f(x) = ½·d/dx[f(x)²]. So:</p><p>∫₋₁⁰ x f'(x)f(x) dx = ½∫₋₁⁰ x · d/dx[f(x)²] dx</p><p><strong>Step 2:</strong> Apply integration by parts with u = x and dv = d/dx[f(x)²]dx:</p><p>½∫₋₁⁰ x · d/dx[f(x)²] dx = ½[x·f(x)²]₋₁⁰ - ½∫₋₁⁰ 1·f(x)² dx</p><p><strong>Step 3:</strong> Evaluate the boundary term:</p><p>- At x = 0: 0·[f(0)]² = 0</p><p>- At x = -1: -1·[f(-1)]² = -1·(2)² = -4</p><p>So: ½[0 - (-4)] = ½(4) = 2</p><p><strong>Step 4:</strong> Use the given condition ∫₋₁⁰ [f(x)]² dx = 10:</p><p>∫₋₁⁰ x f'(x)f(x) dx = 2 - ½(10) = 2 - 5 = <strong>-3</strong></p><p>∴ Answer: D</p>
Correct Answer: D