February/March 2026 Paper 32

Using the substitution $x=\sqrt{3}\tan u$, show that $I={\int }_{\frac{1}{6}\pi }^{\frac{1}{3}\pi }3{\tan }^{3}u\mathrm{d}u$.

Hence, or otherwise, find the exact value of $I$. Give your answer in the form $p+q\ln r$, where $p$, $q$ and $r$ are rational.