Consider the function $f \in C_{st}$ which on the interval $]-\pi, \pi[$ is equal to the function $x \cos(x)$. Then I have to find the Fourier coefficients $c_{-1}, c_0, c_1$. I know that the Fourier coefficients are defined as $$ c_n = \frac{1}{2 \pi} \int_{-\pi}^{\pi} x \cos(x) e^{-inx} dx $$ and that to calculate $c_1$ for an example I know that I just have to substitute $n=1$ and then calculate the integral but it is a real struggle for me. Do you mind helping me calculating $c_1$ so I can see how to approach questions like this? I know that $c_0 = 0$ as $x \cos(x)$ is an ueven function.
Thank you very much in advance.
Rather than express this in exponential form, express it in trigonometric form:
$\displaystyle b_n = \frac 1 \pi \int_0^\pi x \cos x \sin n x \mathrm d x$
which you can do because it is odd. This is then the half range Fourier sine series.
I will leave it to you to do the work.