Consider the #2\pi#-periodic even function #f# determined by #f(t)=t# for #0\leq t\leq \pi#. The Fourier series of #f# is given by \[s_f(t)={{\pi}\over{2}}+\sum_{n=1}^{\infty}{{2\cdot \left(-1\right)^{n}-2}\over{\pi\cdot n^2}}\cos(n t) \]
Determine the Fourier series #s_F# of the #2\pi#-periodic function whose value is #F(t)= \int_0^t f(\tau)\,\dd \tau# for #t\in \ivco{-\pi}{\pi}# by entering the Fourier coefficients #A_0#, #A_n#, and #B_n# (for #n=1,2,\ldots)# of #s_F# so that
\[s_F(t)=\frac{A_0}{2}+\sum_{n=1}^{\infty}\left(A_n\cos(n t)+B_n\sin(n t)\right) \]
Simplify your answer so it contains no trigonometric functions.