Pythagorean identities will just have to do

Calculus Level 5

\[ \large \mathscr{E} = \displaystyle \int_{0}^{\pi/2} \sqrt{1- \sin(2016 x) }\, dx \]

Find \( \left \lfloor 1000 \mathscr{E} \right \rfloor \).

Bonus: Make and prove generalization for \( \displaystyle \int_{0}^{\pi/2} \sqrt{1- \sin(12n x) }\, dx \), where \(n\) is a positive integer.


Inspiration.

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