Antiderivative of periodic function

Hint(s):

Start with the definition of the antiderivative(s) of $f$:

$$F(x) = \int_0^xf(t)dt + C$$

where $C$ is a constant. Since $f$ is periodic, with period $T>0$, then

$$F(2T) = \int_0^{2T}f(t)dt + C = \int_0^{T}f(t)dt + \int_T^{2T}f(t)dt + C = 2\int_0^{T}f(t)dt + C$$ $$F(3T) = \int_0^{3T}f(t)dt + C = F(2T) + \int_T^{2T}f(t)dt = 3\int_0^{T}f(t)dt + C$$

Can you generalize this to find an expression for $F(kT)$ for any integer $k$ (for example by induction)? What does this tell you about $\int_0^{T}f(t)dt$ given the conditions $F$ have to satisfy?

Finally what is the value of

$$F(x+T) - F(x) = \int_x^{x+T}f(t)dt = \int_x^{T}f(t)dt + \int_T^{x+T}f(t)dt$$

given the result found above?