Coziness Costs

Calculus Level 5

\[\displaystyle \int_0^{\pi/6} {\sum_{n=1}^{\infty}{\left(\cos(x) \cos\left(x + \frac{\pi}{n}\right) \cos\left(x + \frac{2\pi}{n}\right) \cdots \cos\left(x + \frac{(n-1)\pi}{n}\right)\right)}\, dx} = \ln\left(\dfrac{a}{b}\right)\]

Find \(ab\).

Details and Assumptions

\(a,b\) are coprime positive integers.

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