Fourier series of a system

Question

I have a system: f(x) = $ \begin{cases} \sin x , x \in [0, \pi);\\ 0, x \in [\pi,2\pi].\end{cases}$

Am I right, that I just need to consider a sum of integrals in all formulas?

And period is still $2\pi$ so l = $\pi$.

Answer 1

$$\hat f(n) = \frac{1}{2\pi}\int_{0}^{2\pi} f(x)e^{-inx} dx = \frac{1} {2\pi}\int_{0}^{\pi}sin(x)e^{-inx} dx$$

because $f = 0$ for $x \in [\pi,2\pi]$.

Now write $sin(x) = \frac{e^{ix}-e^{-ix}}{2i}$ and integrate simply.