How to integrate this function:

$$x[n] = \frac{1}{2\pi}\int\_{-\pi}^{\pi}\delta(\omega-\omega\_{0}) e^{jn\omega}d\omega$$

Where: 

$$\delta[\omega-\omega\_0] = \begin{cases}1&n=\omega\_o\\0&n\neq\omega\_o\end{cases}$$

The book says the answer is suppose to be:

$$x[n] = \frac{1}{2\pi}e^{jn\omega\_o} $$

Here's what i'm trying to do:

$$x[n] = \frac{1}{2\pi}\int\_{-\pi}^{\pi}\delta(\omega-\omega\_{0}) e^{jn\omega}d\omega$$

$$x[n] = \frac{1}{2\pi} e^{jn\omega\_o} \int\_{-\pi}^{\pi}d\omega$$

$$x[n] = \frac{1}{2\pi} e^{jn\omega\_o} (\pi - (-\pi))$$

$$x[n] = \frac{1}{2\pi} e^{jn\omega\_o} (2\pi)$$

$$x[n] = e^{jn\omega\_o} $$