# Trigonometry and Calculus!

Calculus Level 5

$\large F(x)=\dfrac{\sin x\cos x}{\sqrt{19\sin^2 x+10\cos^2 x+6}}$

Let $$0< \alpha < \dfrac \pi 2$$ such that $$\max(F(x)) = F(\alpha) = \phi$$.

Given that the value of $\displaystyle \large \mathcal H = \int_{\pi (9\phi - 1)}^\alpha F(x) \, dx$ is equal to $$\dfrac AB(\sqrt C - D)$$, where $$A,B,C$$ and $$D$$ are positive integers with $$A,B$$ coprime and $$C$$ square-free.

Find $$A+B+C+D$$.

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