Đáp án:

• Đặt \( t = \sqrt{1 + 3 \ln x} \Rightarrow \ln x = \frac{1}{3}(t^2 - 1) \Rightarrow \frac{1}{x} dx = \frac{2}{3} t \, dt \)

• \( \int \frac{\ln x \sqrt{1 + 3 \ln x}}{x} \, dx = \int \frac{1}{3} (t^2 - 1) t \cdot \frac{2}{3} t \, dt \)

\( = \frac{2}{9} \int (t^4 - t^2) \, dt \)

\( = \frac{2}{9} \left[ \frac{t^5}{5} - \frac{t^3}{3} \right] + C \)