Question

If \( \int x^x (1 + \log x) \, dx = k x^x + c \), then \( k = \)

• A. \( \log_e e \)
• B. \( \log_e \left( \dfrac{1}{e^2} \right) \)
• C. \( \log_e (e^2) \)
• D. \( \log_e \left( \dfrac{1}{e} \right) \)

Hint:

Always remember: \( \dfrac{d}{dx}(x^x) = x^x(1+\log x) \).

Answer 1

The Correct Option is A

Step 1: Identify the derivative.
We know that \[ \frac{d}{dx}(x^x) = x^x (1 + \log x) \] Step 2: Compare with the given integral.
\[ \int x^x (1 + \log x) \, dx = x^x + c \] Step 3: Match coefficients.
Comparing with \( k x^x + c \), we get \[ k = 1 = \log_e e \] Step 4: Conclusion.
Hence, \( k = \log_e e \).