A Sinful Sinfonia!

Calculus Level 5

0π/3n=1[sin(x)sin(x+πn)sin(x+2πn)sin(x+(n1)πn)]dx=lna\displaystyle \int_0^{\pi/3} {\sum_{n=1}^{\infty}{\left[\sin(x) \sin\left(x + \frac{\pi}{n}\right) \sin\left(x + \frac{2\pi}{n}\right) \cdots \sin\left(x + \frac{(n-1)\pi}{n}\right)\right]}\, dx} = \ln a

Find aa.

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