# 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?

Note by Timothy Wan
2 years, 4 months ago

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Hint: Start with the substitution $$x = e^{-y}$$. Read Gamma function

- 2 years, 3 months ago