Challenging Integral!

Calculus Level 3

Let \(\displaystyle I(n) = \int_1^e x^3 (\ln x)^n \, dx \), where \(n\) is a positive integer, find the value of \( \dfrac{4I(n) + nI(n-1)}{e^4} \) for \(n\geq1\).

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