Definite Integration
Properties of Definite Integrals
Grade 12
Question:
<p>If \(f(x+y) = f(x)f(y)\) for all \(x, y\) and \(f(0) \neq 0\), and \(F(x) = \frac{f(x)}{1 + (f(x))^2}\), then:</p>
<p>(a) \(\int_{-2010}^{2011} F(x) dx = \int_0^{2011} F(x) dx\)</p>
<p>(b) \(\int_{-2010}^{2011} F(x) dx - \int_0^{2010} F(x) dx = \int_0^{2011} F(x) dx\)</p>
<p>(c) \(\int_{-2010}^{2011} F(x) dx = 0\)</p>
<p>(d) \(\int_0^{2010} (2F(x) - ...)\)</p>
Step-by-Step Solution
Key Concept: Recognize exponential functions from functional equations and use symmetry properties.
<p>The functional equation $f(x+y) = f(x)f(y)$ implies $f(x) = e^{cx}$ for some constant $c$. Then $F(x) = \frac{e^{cx}}{1 + e^{2cx}}$ is an odd function when shifted appropriately, leading to symmetry properties in definite integrals.</p>
Correct Answer: a