Indefinite Integration
Recognition Method
Grade 12
Question:
<p><span class="math">\(\int x^x(1 + \log|x|) \, dx\)</span> is equal to</p>
<p>(a) <span class="math">\(x^x \log|x| + C\)</span></p>
<p>(b) <span class="math">\(e^{x} x + C\)</span></p>
<p>(c) <span class="math">\(x^x + C\)</span></p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Recognize that the integrand is the exact derivative of x^x. Use the derivative formula for exponential-power functions.
<p><strong>Solution:</strong> Note that <span class="math">$\frac{d}{dx}(x^x) = x^x(\log x + 1) = x^x(1 + \log|x|)$</span>. Therefore <span class="math">$\int x^x(1 + \log|x|) \, dx = x^x + C$</span>.</p>
Correct Answer: C