They are using two different "normalizations" because they are calculating the Fourier series over two different intervals.

Remember that youcannotactually find a Fourier series for "f(x)= x". You can only find Fourier series forperiodicfunctions.

In the first case they are finding the Fourier series for the function that is equal to x for and then repeats that value with period .

In the second case they are finding the Fourier series for the function that is equal to x for and then repeats that value with period 2.