Fourier Series of Triangular waveform

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Fourier Series of Triangular waveform

this is the solution of Fourier series of a triangular waveform from the book Circuits and Networks: Analysis and Synthesis by Shyammohan S. Palli. In this problem they have take the time period of the triangular waveform from -π to +π instead of 0 to 2π.

Now, from -π to 0 the equation of the waveform is as shown below

and from 0 to +π the equation of the waveform is

Which gives

My question is,

If i take the interval from 0 to 2π.then from 0 to +π the equation of the waveform will be

and from +π to +2π the equation of the waveform will be

But this gives

which did not match with previous.My question is why they took the interval from -π to +π instead of 0 to 2π.

I will post my calculation here

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