I am a bit confused by the complex Fourier Series in general, but I am working with complex Fourier Series on the form: $$\sum_{n=-\infty}^{\infty} c_ne^{inx}$$
Where the Fourier coefficient $c_n$ is defined as: $$ c_n = \frac{1}{2\pi} \int_{-\pi}^{\pi}f(y)e^{-iny}dy$$
I am then find the $c_0$ and the $c_n$ coefficient of the function on the interval $]-\pi, \pi]$ defined by:
$$f(x) := \begin{cases} 0, & \text{for $-\pi<x<0$} \\ sin(x), & \text{for $0\le x\le \pi $} \end{cases}$$
This is what I have done so far:
Note that $sin(x) = \frac{e^{ix} - e^-{ix}}{2i}$, so we have that
$$c_n = \frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx}dx=\frac{1}{2\pi}\int_{-\pi}^{0}0(x)e^{-inx}dx+\frac{1}{2\pi}\int_{0}^{\pi}sin(x)e^{-inx}dx = \frac{1}{2\pi}\int_{0}^{\pi}(\frac{e^{ix} - e^{-ix}}{2i})e^{-inx}dx$$
What am I supposed to do here in order to find $c_0$ and $c_n$ ?