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Sophomore’s dream constant

Evaluate the integral

    \[\mathcal{J} = \int_0^1 \int_0^1 (xy)^{xy} \,\mathrm{d}(x, y)\]

Solution

Let t=xy and s=y. The Jacobian is

    \[\frac{\partial (s, t)}{\partial (x, y)} = y \Rightarrow \left (\frac{\partial (s, t)}{\partial (x, y)} \right )^{-1} = \frac{1}{y} = \frac{1}{s}\]

Hence,

    \begin{align*} \mathcal{J} &= \iint \limits_{[0, 1]^2} \left ( xy \right )^{xy} \, \mathrm{d}(x, y) \\ &=\int_{0}^{1} \int_{0}^{s} \frac{t^t}{s} \, \mathrm{d}(t, s) \\ &= \int_{0}^{1} \int_{t}^{1} \frac{t^t}{s} \, \mathrm{d} ( s, t)\\ &= -\int_{0}^{1} t^t \log t \, \mathrm{d}t \end{align*}

However , since \displaystyle \int_{0}^{1} t^t \left ( 1 + \log t \right ) \, \mathrm{d}t =0 we conclude that

    \[\mathcal{J} = \int_0^1 t^t \, \mathrm{d}t = \mathcal{S}\]

where \mathcal{S} is Sophomore’s dream constant.

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