Product Rule & Integration by Parts

Review question

1. Prove that $\begin{equation*} \int_0^{\frac{1}{2}\pi} \frac{\cos x}{3 + \cos^2 x} \:dx = \tfrac{1}{4} \ln 3. \end{equation*}$
2. Integrate with respect to $x$ $\begin{equation*} (a)\; \frac{x}{(x-1)^3}, \quad (b)\;(\ln x)^2. \end{equation*}$