Arctan and tan

Calculus Level 4

\[\large I = \sum\limits_{\alpha=2} ^{2047} \left( \int\limits_{0}^{\pi/2} \frac{\arctan \left( \frac{ \tan(x)}{\alpha}\right)}{\tan(x)}dx\right) \]

Given the expression above, find \(\exp \left(\frac{I}{\pi} \right)\).

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