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Calculus Level 4

\[\begin{align} \lambda&=\int_{0}^{1}\frac{dx}{1+x^{3}} \\ p&=\lim_{n\to \infty} \left[{\frac{\prod_{r=1}^{n}(n^{3}+r^{3})}{n^{3n}}}\right] ^{\frac{1}{n}}\end {align} \] If \(\lambda\) and \(p\) are given as above, find \(\ln(p)\) in terms of \(\lambda\).

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