Ans: (A)
\(\int \frac{\log \sqrt{x}}{3 x} d x=I\)
Let \(\log \sqrt{x}=z \Rightarrow \frac{1}{2 x} d x=d z\)
$$
\begin{aligned}
\therefore I=\int \frac{2 z}{3} d z &=\frac{2}{3} \int z d z \\
&=\frac{2}{3} \cdot \frac{z^{2}}{z}+c \\
&=\frac{1}{3} \cdot(\log \sqrt{x})^{2}+c
\end{aligned}
$$