Definite Integral of Fourier Series at Ends of Interval

Theorem
Let $f: \R \to \R$ be a real function defined in the open interval $\openint {-\pi} \pi$.

Let $f$ fulfil the Dirichlet conditions in $\openint {-\pi} \pi$.

Let $a_0, a_1, \dotsc; b_1, \dotsc$ be the Fourier coefficients of $f$ in $\openint {-\pi} \pi$.

Consider the real function:
 * $\map F x = \ds \int_{-\pi}^x \map f t \rd t - \dfrac {a_0} 2 x$

Then:
 * $\map F \pi = \map F {-\pi} = \dfrac {a_0 \pi} 2$

Proof
From Definite Integral on Zero Interval:

Then: