But I See Two Variables!

Calculus Level 3

\[\large \int_0^{\pi/6}\dfrac{\cos 2x-\cos 2\beta}{\cos x - \cos\beta}\, d\beta=\dfrac{A\pi}B\cos x+A\]

Let \(x\) be a constant real number such that \( \cos x - \cos\beta \ne 0 \) for \( 0 \leq \beta \leq \dfrac\pi 6 \).

If the above equation is true for integers \(A\) and \(B\), then find \(2(A+B)\).

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