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Calculus Level 4

$\large f( x ) =\int _{ 1/e }^{ \tan { x } }{ \dfrac { t }{ 1+{ t }^{ 2 } } \, dt } +\int _{1/e }^{ \cot x }{ \dfrac { 1 }{ t( 1+{ t }^{ 2 } ) } \, dt }$

Find the value of $f\left(\dfrac{e}{2}\right)$.

Clarification: $e \approx 2.71828$ denotes the Euler's number.

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