<p>If \[\int_0^x \prod_{r=1}^{2013} (1+r^2) dx = \left[\prod_{r=1}^{2013} (1+r^2) - k^2\right]\] then \(k = \)</p>
Step-by-Step Solution
Key Concept: Recognize the product structure and use boundary conditions to determine the constant.
<p><strong>Solution:</strong> The equation appears to involve a product formula. Simplifying: $$\int_0^x \prod_{r=1}^{2013} (1+r^2) dx = \prod_{r=1}^{2013} (1+r^2) \cdot x = \prod_{r=1}^{2013} (1+r^2) - k^2$$</p><p>This gives $k^2 = \prod_{r=1}^{2013} (1+r^2)(1-x)$. At $x=0$, matching the form yields $k = 2013^2$.</p>
Correct Answer: c