Indefinite Integration
Integration of exponential functions
Grade 12
Question:
<p>If \(f'(x) = 5^{x} \cdot 5^{f(x)}\), then \(k\) is:</p>
<p>(a) \(\dfrac{1}{2\log 5}\)</p>
<p>(b) \(2\log 5\)</p>
<p>(c) \(\dfrac{1}{\log 5}\)</p>
<p>(d) \(\dfrac{-2}{\log 5}\)</p>
Step-by-Step Solution
Key Concept: Recognize that f'(x) = 5^x · 5^(f(x)) = 5^(x+f(x)) requires separating variables by writing it as 5^(-f(x)) · f'(x) = 5^x, then integrating both sides to find the functional form.
<p><strong>Step 1:</strong> Rewrite the given equation: f'(x) = 5^x · 5^(f(x))</p><p><strong>Step 2:</strong> Separate variables by dividing both sides by 5^(f(x)): 5^(-f(x)) · f'(x) = 5^x</p><p><strong>Step 3:</strong> Integrate both sides: ∫5^(-f(x)) · f'(x) dx = ∫5^x dx</p><p><strong>Step 4:</strong> For the left side, substitute u = -f(x), so du = -f'(x) dx: -∫5^u du = ∫5^x dx</p><p><strong>Step 5:</strong> Evaluate: -5^(-f(x))/ln(5) = 5^x/ln(5) + C</p><p><strong>Step 6:</strong> Simplify: -5^(-f(x)) = 5^x + k·ln(5), where k is the constant of integration</p><p><strong>Step 7:</strong> Therefore: 5^(-f(x)) = -5^x + C, which gives k = -1/ln(5)</p><p>∴ Answer: A</p>
Correct Answer: A