You can make the change of variable x = wt, and work for x in [0, 2*Pi], no problem.
The Fourier series are a bit of a hassle to do. Since your waveform isn't even nor odd, you must consider sin and cos harmonics.
The Fourier series I got is:
\$ \displaystyle \frac{1}{2}\,\sin(x) \ - \frac{1}{\pi} \sum_{n=0}^{\infty}\, \frac{1}{2n+1}\left[\cos\left((4n+1)x\right)-\cos\left((4n+3)x\right)\right] \$
This is a bit more complex than the typical exercise, so I checked it by plotting the Fourier series with n=30, getting the image attached below.
The sine integrals are zero for n>1. The cosine integrals have modulo 4 periodicity, which is a bit ugly.
Edit: Ok, corrected. The first time I got the quarters where the waveform was zero in reverse. Now it's corrected.