Hva er ∫0Lsin2nπxL dx\displaystyle \displaystyle\int_0^L \sin^2\frac{n\pi x}{L}\,dx∫0Lsin2Lnπxdx?
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∫0Lsin2nπxLdx=∫0L12(1−cos2nπxL)dx=L2\displaystyle \int_0^L\sin^2\frac{n\pi x}{L}dx=\int_0^L\tfrac12\left(1-\cos\frac{2n\pi x}{L}\right)dx=\dfrac{L}{2}∫0Lsin2Lnπxdx=∫0L21(1−cosL2nπx)dx=2L for alle n≥1n\geq 1n≥1. Brukes som nevner i Fourier-koeffisientene.
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