This awesome integral is due to Johann Bernoulli. I think asking to prove it will make a very good project for Calculus II (even Calculus I):
$$\int_0^1\frac{1}{x^x}dx=\frac{1}{1}+\frac{1}{2^2}+\frac{1}{3^3}+\frac{1}{4^4}+\ldots$$
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