Pythagorean identities will just have to do

Calculus Level 3

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

Find \(\mathscr{E}.\)


Inspiration

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

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