2. However, it is fine to take the integral from 0 to 1, since that covers one complete period of this periodic function.
3. it would be helpful if you reserved the summation symbol \sum for sums, and used the integral symbol \int for integrals.
4. The function has period 1, so you should put
5. For the same reason, you should put (because always).
If you do all that, then you get the formula and you can evaluate that integral using integration by parts.