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A not so simple integral

In a certain book, I read that \[ \int_{0}^{1} (x \ln x)^k \ dx = \frac{(-1)^k k!}{(k+1)^{k+1}} \forall \mathbb{Z}_{\geq 0 } \] However, it does not state, or explain how it was derived.

How does one solve for the integral of \( (x \ln x)^k \) and how was the answer derived?

Please help, and thanks in advance. - Timothy

Note by Timothy Wan
1 year, 6 months ago

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Hint: Start with the substitution \(x = e^{-y}\). Read Gamma function Pi Han Goh · 1 year, 6 months ago

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