By definition of Fourier series: :$\displaystyle \map f x \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \cos n \pi x + b_n \sin n \pi x}$ where: {{begin-eqn}} {{eqn | l = a_n       | r = \int_0^2 \map f x \cos n \pi x \rd x }} {{eqn | l = b_n       | r = \int_0^2 \map f x \sin n \pi x \rd x }} {{end-eqn}} for all $n \in \Z_{>0}$. Thus: {{begin-eqn}} {{eqn | l = a_0       | r = \int_0^2 \map f x \rd x       | c = Cosine of Zero is One }} {{eqn | r = \int_0^2 \paren {4 - x^2} \rd x       | c =  }} {{eqn | r = \intlimits {4 x - \frac {x^3} 3} 0 2       | c = Primitive of Power }} {{eqn | r = \paren {4 \times 2 - \frac {2^3} 3} - \paren {0 - 0}       | c =  }} {{eqn | r = 8 - \frac 8 3       | c =  }} {{eqn | r = \frac {16} 3       | c =  }} {{end-eqn}} {{qed|lemma}} For $n > 0$: {{begin-eqn}} {{eqn | l = a_n       | r = \int_0^2 \map f x \cos n \pi x \rd x       | c =  }} {{eqn | r = \int_0^2 \paren {4 - x^2} \cos n \pi x \rd x       | c = Definition of $f$ }} {{eqn | r = 4 \int_0^2 \cos n \pi x \rd x - \int_0^2 x^2 \cos n \pi x \rd x       | c = Linear Combination of Integrals }} {{end-eqn}} Splitting this up into bits: {{begin-eqn}} {{eqn | o =        | r = 4 \int_0^2 \cos n \pi x \rd x       | c =  }} {{eqn | r = 4 \intlimits {\frac {\sin n \pi x} {n \pi} } 0 2       | c = Primitive of $\cos n \pi x$ }} {{eqn | r = 4 \paren {\frac {\sin 2 n \pi} {n \pi} } - 4 \paren {\frac {\sin 0} {n \pi} }       | c =  }} {{eqn | r = 0       | c = Sine of Multiple of Pi }} {{end-eqn}} {{begin-eqn}} {{eqn | o =        | r = \int_0^2 x^2 \cos n \pi x \rd x       | c =  }} {{eqn | r = \intlimits {\frac {2 x \cos n \pi x} {\paren {n \pi}^2} + \paren {\frac {x^2} {n \pi} - \frac 2 {\paren {n \pi}^3} } \sin n \pi x} 0 2       | c = Primitive of $x^2 \cos n \pi x$ }} {{eqn | r = \paren {\frac {4 \cos 2 n \pi} {n^2 \pi^2} + \paren {\frac 4 {n \pi} - \frac 2 {\paren {n \pi}^3} } \sin 2 n \pi} - \paren {\frac {0 \cos 0} {n^2 \pi^2} + \paren {\frac 0 {n \pi} - \frac 2 {\paren {n \pi}^3} } \sin 0}       | c =  }} {{eqn | r = \frac {4 \cos 2 n \pi} {n^2 \pi^2}       | c = Sine of Multiple of Pi and removal of vanishing terms }} {{eqn | r = \frac 4 {n^2 \pi^2}       | c = Cosine of Multiple of Pi }} {{end-eqn}} Reassembling $a_n$ from the remaining non-vanishing terms: {{begin-eqn}} {{eqn | l = a_n       | r = 0 - \frac 4 {n^2 \pi^2}       | c =  }} {{eqn | r = -\frac 4 {n^2 \pi^2}       | c =  }} {{end-eqn}} {{qed|lemma}} Now for the $\sin n \pi x$ terms: {{begin-eqn}} {{eqn | l = b_n       | r = \int_0^2 \map f x \sin n \pi x \rd x       | c =  }} {{eqn | r = \int_0^2 \paren {4 - x^2} \sin n \pi x \rd x       | c = Definition of $f$ }} {{eqn | r = 4 \int_0^2 \sin n \pi x \rd x - \int_0^2 x^2 \sin n \pi x \rd x       | c = Linear Combination of Integrals }} {{end-eqn}} Splitting this up into bits: {{begin-eqn}} {{eqn | o =        | r = 4 \int_0^2 \sin n \pi x \rd x       | c =  }} {{eqn | r = 4 \intlimits {\frac {-\cos n \pi x} {n \pi} } 0 2       | c = Primitive of $\sin n \pi x$ }} {{eqn | r = 4 \paren {\frac {-\cos 2 n \pi} {n \pi} } - 4 \paren {\frac {-\cos 0} {n \pi} }       | c =  }} {{eqn | r = 4 \paren {\frac {-1} {n \pi} } - 4 \paren {\frac {-1} {n \pi} }       | c = Cosine of Multiple of Pi }} {{eqn | r = 0       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | o =        | r = \int_0^2 x^2 \sin n x \rd x       | c =  }} {{eqn | r = \intlimits {\frac {2 x \sin n \pi x} {\paren {n \pi}^2} + \paren {\frac 2 {\paren {n \pi}^3} - \frac {x^2} {n \pi} } \cos n \pi x} 0 2       | c = Primitive of $x^2 \sin n \pi x$ }} {{eqn | r = \paren {\frac {4 \sin 2 n \pi} {n^2 \pi^2} + \paren {\frac 2 {\paren {n \pi}^3} - \frac 4 {n \pi} } \cos 2 n \pi} - \paren {\frac {0 \sin 0} {n^2 \pi^2} + \paren {\frac 2 {\paren {n \pi}^3} - \frac 0 {n \pi} } \cos 0}       | c =  }} {{eqn | r = \paren {\frac 2 {\paren {n \pi}^3} - \frac 4 {n \pi} } \cos 2 n \pi - \frac 2 {\paren {n \pi}^3} \cos 0       | c = Sine of Multiple of Pi and removal of vanishing terms }} {{eqn | r = -\frac 4 {n \pi}       | c = Cosine of Multiple of Pi and simplifying }} {{end-eqn}} Reassembling $b_n$ from the remaining non-vanishing terms: {{begin-eqn}} {{eqn | l = b_n       | r = 0 - \paren {-\frac 4 {n \pi} }       | c =  }} {{eqn | r = \frac 4 {n \pi}       | c =  }} {{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 {16} 3} + \sum_{n \mathop = 1}^\infty \paren {-\frac 4 {n^2 \pi^2} \cos n \pi x + \frac 4 {n \pi} \sin n \pi x}       | c = substituting for $a_0$, $a_n$ and $b_n$ from above }} {{eqn | r = \frac 8 3 - \frac 4 {\pi^2} \sum_{n \mathop = 1}^\infty \frac {\cos n \pi x} {n^2} + \frac 4 \pi \sum_{n \mathop = 1}^\infty \frac {\sin n \pi x} n       | c = rearranging }} {{end-eqn}} {{qed}}