Integrate with sin cos

Calculus Level 5

$\large\displaystyle \int_{0}^{\frac{\pi}{4}} (\cos 2 \theta)^{\frac{3}{2}} \cos \theta \, d \theta$

Given the integral above equals to $\dfrac{A}{B \sqrt{C}} \pi$ where $A,B,C$ are positive integers with $A,B$ are coprime and $C$ square-free. Find $B-A-C$ .

Integrate with sin cos

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