Definite Integration
King's Rule / Symmetry properties
Grade 12
Question:
<p>Let <em>f</em> and <em>g</em> be continuous functions on <em>[-20, 20]</em> such that <em>f(-x) = f(x)</em>, <em>g(-x) = -g(x)</em>, and <em>g(x) = 1 + f(x)·f(-x)</em>. If \(\int_{-20}^{20} f(x)\,dx = 2020\), then \(I = \int_{-20}^{20} \dfrac{f(x)}{g(x)}\,dx\) equals:</p>
<p>2020</p>
<p>1010</p>
<p>505</p>
<p>0</p>
Step-by-Step Solution
Key Concept: Use the property that f is even and g is odd to split the integrand. Since f(-x)=f(x) is even and g(-x)=-g(x) is odd, the product f(x)/g(x) is odd (even/odd = odd), making its integral over [-20,20] equal zero.
<p><strong>Step 1: Identify parity of f and g</strong></p><p>Given: f(-x) = f(x) ⟹ f is <strong>even</strong></p><p>Given: g(-x) = -g(x) ⟹ g is <strong>odd</strong></p><p><strong>Step 2: Determine parity of f(x)/g(x)</strong></p><p>Let h(x) = f(x)/g(x)</p><p>h(-x) = f(-x)/g(-x) = f(x)/(-g(x)) = -f(x)/g(x) = -h(x)</p><p>Therefore, h(x) is <strong>odd</strong></p><p><strong>Step 3: Apply odd function integration property</strong></p><p>For any odd function h over a symmetric interval [-a, a]:</p><p>∫₍₋ₐ₎^a h(x)dx = 0</p><p><strong>Step 4: Evaluate the integral</strong></p><p>I = ∫₍₋₂₀₎^20 f(x)/g(x) dx = 0</p><p>∴ Answer: <strong>0</strong></p>
Correct Answer: B