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Product Rule & Integration by Parts

Review question

Can we find all three integrals?
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Ref: R5126

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Question

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