# Trigonometric Integration

Calculus Level 4

$\displaystyle\int_0^{\pi/2}\int_0^{\pi/2}\left(\sqrt{1+\cos(x-y)+\cos(x+y)\cos(x-y)+\cos(x+y)}\right)\mathrm{d}x~\mathrm{d}y$

If the above expression can be simplified to $$\large \color{red}{\alpha}\pi^{\color{purple}{\large\psi}}$$ for some $$\large\color{red}{\alpha},\color{purple}{\psi}\in\mathbb{Z}$$,then:

$\Large(\color{red}{\alpha}+\color{purple}{\psi}+6)^{(\color{red}{\alpha}-\color{purple}{\psi})!}=\ ?$

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