Question:
\(\int x^x \log(ex) dx\) is equal to
\(\int x^x \log(ex) dx\) is equal to
Show Hint
Recognize that \(\frac{d}{dx}(x^x) = x^x(1+\log x)\).
Updated On: Apr 23, 2026
- \(x^x + c\)
- \(x \cdot \log x + c\)
- \((\log x)^x + c\)
- \(x^{\log x} + c\)
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The Correct Option is A
Solution and Explanation
Step 1: Formula / Definition}
\[ x^x = e^{x\log x} \]
Step 2: Calculation / Simplification}
\(\log(ex) = \log e + \log x = 1 + \log x\)
\(\int x^x (1 + \log x) dx\)
Let \(t = x^x \Rightarrow dt = x^x(1 + \log x)dx\)
\(\int dt = t + c = x^x + c\)
Step 3: Final Answer
\[ x^x + c \]
\[ x^x = e^{x\log x} \]
Step 2: Calculation / Simplification}
\(\log(ex) = \log e + \log x = 1 + \log x\)
\(\int x^x (1 + \log x) dx\)
Let \(t = x^x \Rightarrow dt = x^x(1 + \log x)dx\)
\(\int dt = t + c = x^x + c\)
Step 3: Final Answer
\[ x^x + c \]
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