# Prime sine power

Calculus Level 4

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

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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