Indefinite Integration
Exponential Functions
Grade 12
Question:
<p>Evaluate: \(\int x^x \left(\frac{(\ln x)^2}{x} + \frac{\ln x}{x} + \frac{1}{x}\right) dx\)</p>
<p>(a) \(x^x \left((\ln x)^2 - \frac{1}{x}\right) + C\)</p>
<p>(b) \(x^x(\ln x - x) + C\)</p>
<p>(c) \(x^x \frac{(\ln x)^2}{2} + C\)</p>
<p>(d) \(x^x \ln x + C\)</p>
Step-by-Step Solution
Key Concept: The derivative of $x^x f(x)$ is $x^x(f'(x) + f(x)\ln x)$; apply this pattern recognition.
<p>Recognize that the integrand is the derivative of $x^x(\ln x)^2$. Use the product rule and chain rule for $x^x$ to verify the answer.</p>
Correct Answer: A