Evaluation of the Definite Integral $\int_{\zeta=0}^{2x} \lvert \sin(\zeta)\rvert \mathrm{d}\zeta$ I need help understanding how to evaluate the following integral:
$$\int_{\zeta=0}^{2x} \lvert \sin(\zeta)\rvert \mathrm{d}\zeta$$
I am a bit lost so if someone could help out that would be great.

Sketch the graph of $|\sin x|$. Suppose $x>0$. Let $k=\lfloor 2\,x/\pi\rfloor$ be the unique non-negative integer such that $k\,\pi\le2\,x<(k+1)\pi$. Then
$$
\int_0^{2x}|\sin\zeta|\,d\zeta=\int_0^{k\pi}|\sin\zeta|\,d\zeta+\int_{k\pi}^{2x}|\sin\zeta|\,d\zeta=k\int_0^\pi\sin\zeta\,d\zeta+\int_0^{2x-k\pi}\sin\zeta\,d\zeta.
$$
