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**Orthogonal Properties for Sine Don't Hold if Pi is involded??**

Normally I know [tex]

\int_{-L}^L \sin \frac{n x}{L} \sin \frac{\m x}{L} ~ dx = 0\mbox{ if }n\not =m , \ =L \mbox{ if }n=m

[/tex] but apparently this doesn't work for [tex]

\int_{-L}^L \sin \frac{\pi n x}{L} \sin \frac{\pi m x}{L} ~ dx

[/tex]

I am trying to find Fourier coefficients, and my integral is [tex]\int_{0}^{L/2} \sin \frac{2\pi x}{L} \sin \frac{\pi n x}{L} ~ dx

[/tex]

For even n that aren't 2, the integral is 0, if n=2, then the integral is L/2, but if n is odd, then the integral doesn't equal 0 (it's actually a fairly complex answer) even thought it looks like it should by orthogonal properties ...why doesn't this work?

The integrals I am doing take very long to do, is there a property that would allow me to shorten them?

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