Mind explosion!

Calculus Level 2

\[\large \sqrt{\color{red}{a}} ~=~ \sqrt { 2 } \sqrt { \int _{ 0 }^{ \infty }{ \frac { 1 }{ 1+{ t }^{ 2 } } } dt } \]

\[\huge \color{red}{a}^{\log _{ \pi }{ 2 } } ~=~?\]

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