Don't ask why - 17

Calculus Level 5

\[\lim_{p\to0+}I_{17}(p) =\lim_{p\to0+} \int_0^\infty \int_0^\infty \cdots \int_0^\infty \frac{\prod_{j=1}^{17} \cos x_j}{x_1 + x_2 + \cdots + x_{17}}\,e^{-p(x_1+x_2+\cdots + x_{17})}\,dx_1\,dx_2\,\ldots\,dx_{17} \]

If the limit is equal to \(\dfrac{1}{A}\), find \(A\).

Bonus: Find the closed form of

\[\lim_{p\to0+} I_n(p) \; = \; \lim_{p\to0+} \int_0^\infty \int_0^\infty \cdots \int_0^\infty \frac{\prod_{j=1}^n \cos x_j}{x_1 + x_2 + \cdots + x_n}\,e^{-p(x_1+x_2+\cdots + x_n)}\,dx_1\,dx_2\,\ldots\,dx_n \]

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