Indefinite Integration
Properties of Integrated Functions
Grade 12
Question:
<p>Let <span class="math">f(x) = \int x^2 \cos 2x (2x + 6\tan x - 2x\tan^2 x) dx</span> and <span class="math">f(x)</span> passes through the point <span class="math">(\pi, 0)</span>.</p><p>If <span class="math">f: \mathbb{R} - \{(2n+1)\frac{\pi}{2}\} \to \mathbb{R}</span> then <span class="math">f(x)</span> be a:</p>
<p>(a) even function</p>
<p>(b) odd function</p>
<p>(c) neither even nor odd</p>
<p>(d) even as well as odd both</p>
Step-by-Step Solution
Key Concept: Compute the indefinite integral and use the boundary condition to find the constant, then check parity of the resulting function.
<p><strong>Solution:</strong> Integrate the given expression to find <span class="math">f(x)</span>. Using the condition that <span class="math">f(\pi) = 0</span> to determine the constant of integration, then verify whether <span class="math">f(-x) = f(x)</span> (even) or <span class="math">f(-x) = -f(x)</span> (odd).</p>
Correct Answer: a