$$
To simplify the expression, we can rewrite $\sqrt{x}$ and $\ln \sqrt{x}$ in terms of $x$. We know that $\sqrt{x} = x^{1/2}$ and $\ln \sqrt{x} = \ln(x^{1/2}) = \frac{1}{2} \ln x$. Thus, we can rewrite the integral as:
$$\int \sqrt{x} \ln \sqrt{x} \, dx = \int
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