Indefinite Integration
Standard Forms with Logarithms
Grade None
Question:
<p>Evaluate: \(\int \left( a^x \ln x + \ln a \cdot \ln\left(\frac{x}{e}\right) \right) dx\)</p>
<p>(a) \(a^x \ln\left(\frac{x}{e}\right) + C\)</p>
<p>(b) \(a^x \ln\left(\frac{x}{e}\right) + C\)</p>
<p>(c) \(a^x + \ln\left(\frac{x}{e}\right) + C\)</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Recognize that the derivative of $a^x \ln\left(\frac{x}{e}\right)$ yields the given integrand when using the product rule.
<p>This is an indefinite integration problem. The integrand contains $a^x$ and logarithmic terms that suggest the answer involves $a^x$ multiplied by a logarithmic expression.</p>
Correct Answer: A