[tex]\frac{a_{0}}{2} \sum a_{n}cos \frac{nx}{L} b_{n}sin \frac{nx}{L}[/tex]

[tex]a_{0}=\frac{1}{L }\int_{-L}^{L} f(x)[/tex]

[tex]a_{n}=\frac{1}{L }\int_{-L}^{L} f(x)cos \frac{nx}{L}[/tex]

[tex]b_{n}=\frac{1}{L }\int_{-L}^{L} f(x)sin \frac{nx}{L}[/tex]

[tex]a_{0}=\frac{1}{\pi }\int_{-\pi}^{\pi} f(x)[/tex]

[tex]a_{n}=\frac{1}{\pi }\int_{-\pi}^{\pi} f(x)cos nx[/tex]

[tex]b_{n}=\frac{1}{\pi }\int_{-\pi}^{\pi} f(x)sin nx[/tex]

[tex]b_{n}=\frac{2}{\pi }\int_{0}^{\pi} x.sin nx = \frac{2}{\pi }\left ( \frac{-x.cos nx}{n} \frac{sin nx}{n^{2}} \right )_{0}^{\pi}[/tex]

[tex]b_{n}=\frac{2}{\pi }\left ( \left [ \frac{-\pi.cosn\pi}{n} \frac{sinn\pi}{n^{2}} \right ]-\left [\frac{0.cosn0}{n} \frac{sinn0}{n^{2}} \right ] \right )[/tex]

[tex]b_{n}=\frac{-2}{n }cosn\pi[/tex]

[tex]\frac{a_{0}}{2} \sum a_{n}cos nx b_{n}sin nx=2\sum \frac{(-1)^{n 1}}{n}sin nx[/tex]