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Evaluate

$\frac{1}{\pi} \sum _{ n=2 }^{ 101 }{ \int _{ 1 }^{ n }{ \left(2\tan ^{ -1 }{ (x) } +\sin ^{ -1 }{ \left(\frac { 2x }{ 1+x^{ 2 } } \right)}\right)dx } } .$

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