Determine the Fourier coefficients of $$f:(-\pi,\pi]\to\mathbb{R},f(x)=\begin{cases}0,&x<0,\\x,&x\geq 0.\end{cases}$$
My attempt thus far is the following: $$\begin{align*} \frac{1}{2\pi}\int_0^{\pi}x\exp(-\mathrm ikx)\,\mathrm dx &= \frac{1}{2\pi}\left(\left[x\cdot\frac{\exp(-\mathrm ikx)}{-\mathrm ik}\right|_{x=0}^{x=\pi} - \int_0^{\pi}\frac{\exp(-\mathrm ikx)}{-\mathrm ik}\,\mathrm dx\right) \\ &= \frac{1}{2\pi}\left(\frac{\pi}{-\mathrm ik}\exp(-\mathrm ik\pi) - \int_0^{\pi}\frac{\exp(-\mathrm ikx)}{-\mathrm ik}\,\mathrm dx\right) \\ &= \frac{1}{2\pi}\left(\frac{\pi}{-\mathrm ik}\left(\cos(-k\pi)+\mathrm i\sin(-k\pi)\right) - \int_0^{\pi}\frac{\exp(-\mathrm ikx)}{-\mathrm ik}\,\mathrm dx\right) \\ &= \frac{1}{2\pi}\left(\frac{\pi}{-\mathrm ik}(-1)^{k} - \int_0^{\pi}\frac{\exp(-\mathrm ikx)}{-\mathrm ik}\,\mathrm dx\right) \\ &= \frac{1}{2\pi}\left(\frac{(-1)^{k}\mathrm i\pi}{k} - \int_0^{\pi}\frac{\exp(-\mathrm ikx)}{-\mathrm ik}\,\mathrm dx\right) \\ &= \frac{1}{2\pi}\left(\frac{(-1)^{k}\mathrm i\pi}{k} - \left(-\frac{1}{\mathrm ik}\left[\frac{\exp(-\mathrm ikx)}{-\mathrm ik}\right|_{x=0}^{x=\pi}\right)\right) \\ &= \frac{1}{2\pi}\left(\frac{(-1)^{k}\mathrm i\pi}{k} - \left(-\frac{1}{k^2}\left(\exp(-\mathrm ik\pi)-1\right)\right)\right) \\ &= \frac{1}{2\pi}\left(\frac{(-1)^{k}\mathrm i\pi}{k} - \left(-\frac{1}{k^2}\left(\cos(-k\pi)+\mathrm i\sin(-k\pi)-1\right)\right)\right) \\ &= \frac{1}{2\pi}\left(\frac{(-1)^{k}\mathrm i\pi}{k} - \left(-\frac{1}{k^2}\left((-1)^{k}-1\right)\right)\right) \\ &= \frac{1}{2\pi}\left(\frac{(-1)^{k}\mathrm i\pi}{k} - \left(-\frac{(-1)^k}{k^2}+\frac{1}{k^2}\right)\right) \\ &= \frac{1}{2\pi}\left(\frac{(-1)^k\mathrm ik\pi+(-1)^k-1}{k^2}\right) \\ &= \frac{(-1)^k(\mathrm ik\pi+1)-1}{2\pi k^2}. \end{align*}$$
For $k=0$ I just obtain $$\frac{1}{2\pi}\int_0^{\pi}x\,\mathrm dx=\frac{\pi}{4}.$$
I'd like to know whether there are any objections or simplifications I missed.
It looks right to me, a way to make sure you're on the right track is just plotting
$$ f_N(x) =\sum_{k=-N}^N c_k e^{ik x} $$
for some values of $N$ and check that by increasing it the result looks like $f$