By definition of Fourier series: :$\displaystyle \map f x \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \cos n x + b_n \sin n x}$ where: {{begin-eqn}} {{eqn | l = a_n       | r = \dfrac 1 \pi \int_0^{2 \pi} \map f x \cos n x \rd x }} {{eqn | l = b_n       | r = \dfrac 1 \pi \int_0^{2 \pi} \map f x \sin n x \rd x }} {{end-eqn}} for all $n \in \Z_{>0}$. Thus: {{begin-eqn}} {{eqn | l = a_0       | r = \frac 1 \pi \int_0^{2 \pi} \map f x \rd x       | c = Cosine of Zero is One }} {{eqn | r = \frac 1 \pi \int_0^\pi -\pi \rd x + \frac 1 \pi \int_\pi^{2 \pi} \paren {x - \pi} \rd x       | c =  }} {{eqn | r = -\int_0^\pi \rd x + \frac 1 \pi \int_\pi^{2 \pi} x \rd x - \int_\pi^{2 \pi} \rd x       | c = Linear Combination of Integrals }} {{eqn | r = -\int_0^{2 \pi} \rd x + \frac 1 \pi \int_\pi^{2 \pi} x \rd x       | c = Sum of Integrals on Adjacent Intervals for Continuous Functions }} {{eqn | r = -\big[{x}\big]_0^{2 \pi} + \frac 1 \pi \int_\pi^{2 \pi} x \rd x       | c = Primitive of Constant }} {{eqn | r = -\paren {2 \pi - 0} + \frac 1 \pi \int_\pi^{2 \pi} x \rd x       | c =  }} {{eqn | r = -2 \pi + \frac 1 \pi \paren {\intlimits {\frac {x^2} 2} \pi {2 \pi} }       | c = Primitive of Power }} {{eqn | r = -2 \pi + \frac 1 \pi \paren {\frac {\paren {2 \pi}^2} 2 - \frac {\pi^2} 2}       | c =  }} {{eqn | r = -2 \pi + 2 \pi - \frac \pi 2       | c =  }} {{eqn | r = -\frac \pi 2       | c =  }} {{end-eqn}} {{qed|lemma}} For $n > 0$: {{begin-eqn}} {{eqn | l = a_n       | r = \frac 1 \pi \int_0^{2 \pi} \map f x \cos n x \rd x       | c =  }} {{eqn | r = \frac 1 \pi \int_0^\pi \paren {-\pi} \cos n x \rd x + \frac 1 \pi \int_\pi^{2 \pi} \paren {x - \pi} \cos n x \rd x       | c = Definition of $f$ }} {{eqn | r = -\int_0^\pi \cos n x \rd x + \frac 1 \pi \int_\pi^{2 \pi} x \cos n x \rd x - \int_\pi^{2 \pi} \cos n x \rd x       | c = Linear Combination of Integrals }} {{eqn | r = -\int_0^{2 \pi} \cos n x \rd x + \frac 1 \pi \int_\pi^{2 \pi} x \cos n x \rd x       | c = Sum of Integrals on Adjacent Intervals for Continuous Functions }} {{eqn | r = 0 + \frac 1 \pi \int_\pi^{2 \pi} x \cos n x \rd x       | c = Integral over $2 \pi$ of $\cos n x$ }} {{eqn | r = \frac 1 \pi \intlimits {\frac {\cos n x} {n^2} + \frac {x \sin n x} n} \pi {2 \pi}       | c = Primitive of $x \cos n x$ }} {{eqn | r = \frac 1 \pi \paren {\frac {\cos 2 n \pi} {n^2} + \frac {2 \pi \sin 2 n \pi} n} - \frac 1 \pi \paren {\frac {\cos n \pi} {n^2} + \frac {\pi \sin n \pi} n}       | c =  }} {{eqn | r = \frac 1 \pi \paren {\frac {\cos 2 n \pi} {n^2} - \frac {\cos n \pi} {n^2} }       | c = Sine of Multiple of Pi }} {{eqn | r = \frac {1 - \paren {-1}^n} {n^2 \pi}       | c = Cosine of Multiple of Pi and simplifying }} {{eqn | r = \begin {cases} \dfrac 2 {n^2 \pi} & : \text {$n$ odd} \\ 0 & : \text {$n$ even} \end {cases}       | c = separating out the odd and even cases }} {{eqn | r = \dfrac 2 {\pi \paren {2 r + 1}^2}       | c = setting $n = 2 r + 1$ for odd $n$ }} {{end-eqn}} {{qed|lemma}} Now for the $\sin n x$ terms: {{begin-eqn}} {{eqn | l = b_n       | r = \frac 1 \pi \int_0^{2 \pi} \map f x \sin n x \rd x       | c =  }} {{eqn | r = \frac 1 \pi \int_0^\pi \paren {-\pi} \sin n x \rd x + \frac 1 \pi \int_\pi^{2 \pi} \paren {x - \pi} \sin n x \rd x       | c = Definition of $f$ }} {{eqn | r = -\int_0^\pi \sin n x \rd x + \frac 1 \pi \int_\pi^{2 \pi} x \sin n x \rd x - \int_\pi^{2 \pi} \sin n x \rd x       | c = Linear Combination of Integrals }} {{eqn | r = -\int_0^{2 \pi} \sin n x \rd x + \frac 1 \pi \int_\pi^{2 \pi} x \sin n x \rd x       | c = Sum of Integrals on Adjacent Intervals for Continuous Functions }} {{eqn | r = 0 + \frac 1 \pi \int_\pi^{2 \pi} x \sin n x \rd x       | c = Integral over $2 \pi$ of $\sin n x$ }} {{eqn | r = \frac 1 \pi \intlimits {\frac {\sin n x} {n^2} - \frac {x \cos n x} n} \pi {2 \pi}       | c = Primitive of $x \sin n x$ }} {{eqn | r = \frac 1 \pi \paren {\frac {\sin 2 n \pi} {n^2} - \frac {2 \pi \cos 2 n \pi} n} - \frac 1 \pi \paren {\frac {\sin \pi} {n^2} - \frac {\pi \cos \pi} n}       | c =  }} {{eqn | r = \frac {-2 \cos 2 n \pi + \cos n \pi} n       | c = Sine of Multiple of Pi and simplifying }} {{eqn | r = \frac {-2 + \paren {-1}^n} n       | c = Cosine of Multiple of Pi }} {{eqn | r = -\frac {2 - \paren {-1}^n} n       | c = rearranging }} {{end-eqn}} {{qed|lemma}} Finally: {{begin-eqn}} {{eqn | l = \map f x       | o = \sim       | r = \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \cos n x + b_n \sin n x}       | c =  }} {{eqn | r = \dfrac 1 2 \paren {-\frac \pi 2} + \sum_{r \mathop = 0}^\infty \frac 2 {\pi \paren {2 r + 1}^2} \cos \paren {2 r + 1} x + \sum_{n \mathop = 1}^\infty \paren {-\frac {2 - \paren {-1}^n} n} \sin n x       | c = substituting for $a_0$, $a_n$ and $b_n$ from above }} {{eqn | r = -\frac \pi 4 + \frac 2 \pi \sum_{r \mathop = 0}^\infty \frac {\cos \paren {2 r + 1} x} {\paren {2 r + 1}^2} - \sum_{n \mathop = 1}^\infty \frac {2 - \paren {-1}^n \sin n x} n       | c = simplifying }} {{end-eqn}} {{qed}}