Đáp án:

\( f(x) \ln(f(x)) + x f'(x) = x f'(x) f(x) \)

\( \Leftrightarrow \ln(f(x)) + x \cdot \frac{f'(x)}{f(x)} = x f'(x) \)

\( \Leftrightarrow \left( x \ln(f(x)) \right)' = x f'(x) \)

\( \Rightarrow x \ln(f(x)) \big|_{0}^{1} = \int_{0}^{1} x f'(x) dx = x f(x) \big|_{0}^{1} - \int_{0}^{1} f(x) dx \)

\( \Rightarrow \int_{0}^{1} f(x) dx = 1 \Rightarrow \boxed{D} \)