Definite Integration
Functional Equations in Integration
Grade 12
Question:
<p>Suppose <span class="math">f</span> is continuous and satisfies <span class="math">f(x) + f(-x) = x^2</span> then find the value of <span class="math">\int_{-1}^{1} f(x) dx</span></p>
Step-by-Step Solution
Key Concept: Use the given functional equation f(x) + f(-x) = x² along with properties of even and odd functions to establish a relationship that allows integration of f(x) over a symmetric interval.
<p><strong>Step 1:</strong> Given condition: <span class="math">f(x) + f(-x) = x^2</span></p><p><strong>Step 2:</strong> Let <span class="math">I = \int_{-1}^{1} f(x) dx</span></p><p><strong>Step 3:</strong> Using the property that <span class="math">\int_{-a}^{a} f(x) dx = 2\int_{0}^{a} f(x) dx</span> when <span class="math">f(x)</span> is even, and using the given condition:</p><p><span class="math">I = \int_{-1}^{1} f(x) dx = \int_{-1}^{1} \frac{x^2 - f(-x)}{1} dx = \int_{-1}^{1} \frac{x^2}{2} dx = \frac{2}{3}</span></p><p>Note: The property states that <span class="math">\int_{0}^{a} f(a-x) dx = \int_{0}^{a} f(x) dx</span> provided <span class="math">f(a-x) = f(x)</span></p>
Correct Answer: 2/3