Step 1: Understanding the Question:
The question asks for the Integrating Factor (IF) of a linear differential equation.
Step 2: Key Formula or Approach:
For a linear differential equation in standard form $\frac{dy}{dx} + Py = Q$, the integrating factor is $IF = e^{\int P dx}$.
Step 3: Detailed Explanation:
1. Rewrite the given equation in standard form by dividing by $x$:
$\frac{dy}{dx} + (\frac{\log x}{x})y = e^x x^{-1/2} \log x$.
2. Identify $P = \frac{\log x}{x}$.
3. Calculate IF:
$IF = e^{\int \frac{\log x}{x} dx}$.
Using substitution $u = \log x, du = \frac{1}{x} dx$:
$\int \frac{\log x}{x} dx = \int u du = \frac{u^2}{2} = \frac{(\log x)^2}{2}$.
So, $IF = e^{\frac{1}{2} (\log x)^2}$.
4. Let's simplify and check options:
$(\sqrt{x})^{\log x} = (x^{1/2})^{\log x} = x^{\frac{1}{2} \log x} = e^{\ln(x^{\frac{1}{2} \log x})} = e^{\frac{1}{2} \log x \cdot \log x} = e^{\frac{1}{2} (\log x)^2}$.
This matches our calculated result.
Step 4: Final Answer:
The integrating factor is $(\sqrt{x})^{\log x}$.