Painless Intregals

Calculus Level 4

Let \(\displaystyle{ I }_{ a }=\int _{ 0 }^{ { \pi }/{ 2 } }{ \left( \sqrt { \tan { x } } +\sqrt { \cot { x } } \right) dx } \)

and \(\displaystyle { I }_{ b }=\int _{ 0 }^{ \pi }{ \cfrac { x\sin { x } }{ 1+\cos ^{ 2 }{ x } } dx } \).

Find the value of \(\displaystyle \cfrac { { { \left( { I }_{ a } \right) }^{ 2 } } }{ { I }_{ b } } \)

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