Sawtooth Fourier Coefficient

Relevant wiki: Fourier Series

The Fourier coefficients are defined by the integral:

$b_k = \frac{2}{T} \int_0^T f(x) \sin \left(\frac{2 \pi k x}{T} \right)\:dx$

Here we want $k=2$ , with $T=2$ and $f(x) = \frac{x}{2}$ . The integral to compute is thus (using integration by parts):

$b_2 = \int_0^2 \frac{x}{2} \sin(2\pi x) dx = \bigl.-\frac{x}{4\pi} \cos(2\pi x)\bigr|_0^2 + \frac{1}{4\pi}\int_0^2\cos(2\pi x) dx = -\frac{2}{4\pi} = -\frac{1}{2\pi},$

as claimed, since the last integral vanishes.