\[ v(a) = \sum_{i=1}^a \int_0^\pi \sin^{P_i}\left(x P_i\right) \text{d}x \]

Consider the sum of integrals above where \(P_i\) is the \(i\)-th prime number (\(P_1 = 2\)).

\(v\) is of the form \(\dfrac{p}{q} + \dfrac{\pi}{2}\). Give \(q-p\) for \(v(6)\).

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