The constant $a$ satisfies $\int_{1}^{a} x^{2} \ln x \, dx = 4$.
(a)[5]
Show, after rearrangement, that $a = \left( \dfrac{35}{3 \ln a - 1} \right)^{\frac{1}{3}}$.
(b)[2]
Confirm by calculation that $a$ is between $2.4$ and $2.8$.
(c)[3]
Using the iterative formula from part (a), find $a$ correct to $2$ decimal places. Record each iteration to $4$ decimal places.
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