Indefinite Integration
Properties of Integrals
Grade 12
Question:
<p>If \(\int f(x)\,dx = f(x) + C\), then \(\int [f(x)]^2\,dx\) is</p>
<p>(a) \(\frac{1}{2}[f(x)]^2 + C\)</p>
<p>(b) \(\frac{1}{3}[f(x)]^3 + C\)</p>
<p>(c) \(f(x) + C\)</p>
<p>(d) \(\frac{1}{2}[f(x)]^2 + C\)</p>
Step-by-Step Solution
Key Concept: The condition tells us f(x) satisfies a differential equation; use this to evaluate the required integral.
<p>Given \(\int f(x)\,dx = f(x) + C\), this means \(f'(x) = f(x)\), so \(f(x) = e^x\). Then \(\int [e^x]^2\,dx = \int e^{2x}\,dx = \frac{1}{2}e^{2x} + C = \frac{1}{3}[f(x)]^3 + C\).</p>
Correct Answer: B