Complex Fourier Series of $|x|$

Question

How would I write the Fourier series for $|x|$ in complex form over the interval $[-2,2]$? I have already tried writing $$|x|=\sum c_ne^{i\pi nx/2}$$ where \begin{align*}c_n&=\frac{1}{4}\int_{-2}^{2}|x|e^{-i\pi nx/2}dx\\ &=\frac{1}{4}\left(\int_{-2}^0-xe^{-i\pi nx/2}dx+\int_0^2xe^{-i\pi nx/2}dx\right) \end{align*} but that integral seems hopelessly complicated to evaluate. I also have seen this previous post involving $|x|$, but it involves a different interval.

Answer 1

Hint: Integration by parts $$ \int_a^b \underbrace{x}_{=u}\, \underbrace{e^{\lambda x}}_{=v'} \,dx = \bigg(\underbrace{x}_{=u}\,\underbrace{\frac{e^{\lambda x}}{\lambda}}_{=v}\bigg)\Bigg|_a^b - \int_a^b \underbrace{1}_{u'}\,\underbrace{\frac{e^{\lambda x}}{\lambda}}_{=v} \,dx $$