Evaluate an improper logarithmic integral and a revolution volume

(a)(i) Find $\displaystyle\int_{1/3}^{a}\frac{\ln(3x)}{x^3}\,dx$, where $a>1/3$.

(ii) Hence evaluate the same integral with upper limit $\infty$. You may assume $\ln a/a^2\to0$ as $a\to\infty$.

(b) The region under $y=3/(1+4x^2)$ between $x=-\sqrt3/2$ and $x=\sqrt3/2$ is rotated through $2\pi$ about the x-axis. Use $2x=\tan\theta$ to show that the volume is $9\pi\int_0^{\pi/3}\cos^2\theta\,d\theta$, and evaluate it exactly.