Integral Representation of Bessel Function of the First Kind/Non-Integer Order

Theorem
Let $\map {J_n} x$ denote the Bessel function of the first kind of order $n$. Let $n \in \Z$ be an integer.

Then:
 * $\ds \map {J_n} x = \dfrac {x^n} {2^n \sqrt \pi \map \Gamma {n + \frac 1 2} } \int_0^\pi \map \cos {x \sin \theta} \cos^{2 n} \theta \rd \theta$