Evaluate the integral
![\[\mathcal{J} = \int_0^1 \int_0^1 (xy)^{xy} \,\mathrm{d}(x, y)\]](https://www.math.tolaso.com.gr/wp-content/ql-cache/quicklatex.com-ab7b55be97a8d3ef2c1c0f1c3eff9515_l3.png "Rendered by QuickLaTeX.com")
Solution
Let 
and 
. The Jacobian is
![\[\frac{\partial (s, t)}{\partial (x, y)} = y \Rightarrow \left (\frac{\partial (s, t)}{\partial (x, y)} \right )^{-1} = \frac{1}{y} = \frac{1}{s}\]](https://www.math.tolaso.com.gr/wp-content/ql-cache/quicklatex.com-da7407a43114f99b8a61b3c63b05d324_l3.png "Rendered by QuickLaTeX.com")
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*}](https://www.math.tolaso.com.gr/wp-content/ql-cache/quicklatex.com-8113325e4075bd529ea93b94b14170b8_l3.png "Rendered by QuickLaTeX.com")
However , since 
we conclude that
![\[\mathcal{J} = \int_0^1 t^t \, \mathrm{d}t = \mathcal{S}\]](https://www.math.tolaso.com.gr/wp-content/ql-cache/quicklatex.com-01c89222adb8cf39fb6bb79fe5607ac1_l3.png "Rendered by QuickLaTeX.com")
where 
is Sophomore’s dream constant.
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Evaluate the integral
Solution
Let and . The Jacobian is
Hence,
However , since we conclude that
where is Sophomore’s dream constant.