Bessel Function Integral $\int ^{2 \pi}_0 e^{i x \cos t + n t}dt=2\pi i^nJ_n(x),n\in\mathbb{Z}$

Question

$$\int ^{2 \pi}_0 e^{i x \cos t + n t}dt=2\pi i^nJ_n(x),n\in\mathbb{Z}$$

This holds for integer n (although I do not understand why), but what is it equal to if n is not an integer?