Question

\(\int x^x(1 + \log x) \, dx\), \(\text{ is equal to:}\)

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

Hint:

When integrating expressions involving \( x^x \), logarithmic differentiation can simplify the process significantly. Look for opportunities to substitute \( u = x^x \) and use the chain rule for easier evaluation.

Answer 1

The Correct Option is B

We are tasked with evaluating the integral: \[ \int x^x (1 + \log x) \, dx \] Step 1: Use substitution to simplify the integral. Let us first consider the substitution \( u = x^x \), which will simplify the integral. To differentiate \( u = x^x \), we use the logarithmic differentiation technique. \[ \ln u = x \ln x \] Differentiating both sides with respect to \( x \), we get: \[ \frac{1}{u} \frac{du}{dx} = \ln x + 1 \] So, \[ \frac{du}{dx} = u(\ln x + 1) \] This allows us to rewrite the original integral: \[ \int x^x (1 + \log x) \, dx = \int u \, du \] Step 2: Integrate. The integral \( \int u \, du \) is straightforward: \[ \int u \, du = \frac{u^2}{2} + c \] Substitute \( u = x^x \) back into the result: \[ \frac{(x^x)^2}{2} + c = \frac{x^{2x}}{2} + c \] Thus, the integral simplifies to: \[ x^x + c \] Therefore, the correct answer is option (B)