Indefinite Integration
Exponential Functions
Grade 12
Question:
<p>Let \(f(x)\) be a function satisfying \(f'(x) = f(x)\) and \(f(0) = 2\). Then \(\int \frac{f(x)}{3 + 4f(x)} dx\) is</p>
<p>(A) \(\frac{1}{4}\ln(3 + 8e^x) + C\)</p>
<p>(B) \(\frac{1}{8}\ln(3 + 8e^x) + C\)</p>
<p>(C) \(\frac{1}{2}\ln(3 + 8e^x) + C\)</p>
<p>(D) none of these</p>
Step-by-Step Solution
Key Concept: Solve the differential equation to find $f(x)$, then use substitution for the integral.
<p><strong>Step 1:</strong> From $f'(x) = f(x)$ with $f(0) = 2$, we get $f(x) = 2e^x$.</p><p><strong>Step 2:</strong> Substitute into the integral: $\int \frac{2e^x}{3 + 8e^x} dx$</p><p><strong>Step 3:</strong> Let $u = 3 + 8e^x$, then $du = 8e^x dx$</p><p><strong>Step 4:</strong> $\int \frac{2e^x}{3 + 8e^x} dx = \frac{2}{8}\int \frac{du}{u} = \frac{1}{4}\ln|u| + C = \frac{1}{8}\ln(3 + 8e^x) + C$</p>
Correct Answer: B