why is integral of dirchlet from negative pie to pie equal to pie

Question

Hi : I'm reading a text called "Fourier Transformations For Pedestrians " and it's a nice book. But I am stuck on the following. On page 34, the author states that

$\int_{-\pi}^{\pi} D_{N}(x) dx = \pi $ independent of N

where $D_{N}(x)$ represents the Dirichlet kernel and is equal to $1/2 + cos(x) + cos(2x) + \ldots cos(Nx)$.

I don't understand how that is obtained. Thanks for any clarifications.

Answer 1

The integral of all the cosines is $0$ because it ranges over their period an integer number of times.

So the answer is just from the constant term: $${1\over 2}\cdot (\pi-(-\pi))=\pi$$