Tricky BEAST!!!!

Calculus Level 5

Evaluate

$\frac {\displaystyle{ { 2 }^{ 2010 }\int _{ 0 }^{ 1 }{ { x }^{ 1004 } } ({ 1-x) }^{ 1004 }dx} }{\displaystyle{ \int _{ 0 }^{ 1 }{ { x }^{ 1004 } } { (1-{ x }^{ 2010 } })^{ 1004 }dx} }.$

You may try its simpler version first WOWSOME