Bernoulli's identity

In mathematics, sophomore's dream is the pair of identities (especially the first)

discovered in 1697 by Johann Bernoulli.

The name "sophomore's dream", which appears in (Borwein, Bailey & Girgensohn 2004), is in contrast to the name "freshman's dream" which is given to the incorrect identity (x + y)ⁿ = xⁿ + yⁿ. The sophomore's dream has a similar too-good-to-be-true feel, but is true.

We prove the second identity; the first is completely analogous.

The key ingredients of the proof are:

Expand x^x as

Therefore, we have : ${\displaystyle \int _{0}^{1}x^{x}\,\mathrm {d} x=\int _{0}^{1}\sum _{n=0}^{\infty }{\frac {x^{n}(\log x)^{n}}{n!}}\,\mathrm {d} x.}$

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