Question

The value of \( \int x^x(1 + \log x) \, dx \) is equal to ...........

• A. \( \frac{1}{2}(1 + \log x)^2 + c \)
• B. \( x^{2x} + c \)
• C. \( x^x \cdot \log x + c \)
• D. \( x^x + c \)

Hint:

For integrals involving \( x^x \), use the fact that its derivative is \( x^x \cdot (1 + \log x) \).

Answer 1

The Correct Option is A

Step 1: Recall the standard integral.
The given integral is in the form of the derivative of \( x^x \), as: \[ \frac{d}{dx}(x^x) = x^x \cdot (1 + \log x) \]

Step 2: Solve the integral.
Thus, \[ \int x^x(1 + \log x) \, dx = \frac{1}{2} (1 + \log x)^2 + c \]

Step 3: Conclude.
The correct answer is option (i).

Final Answer: \[ \boxed{\frac{1}{2}(1 + \log x)^2 + c} \]