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Thread: inegral with 2 sine fcns

  1. #1
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    inegral with 2 sine fcns

    Hello! I am trying to expand the function $\displaystyle f(x)=sin|x|$ as Fourier series on the interval $\displaystyle |x|<\pi$
    Since the function is even we can write
    $\displaystyle a_{n}=\frac{2}{\pi}\int_{0}^{\pi}sinxcosnxdx$

    but here my problem starts. I try to solve this integral as indefinite integral first but I am going in circles....

    $\displaystyle \int sinxcosnxdx=\{u=cosnx,\ dv=sinx\}=\

    -cosxcosnx-\frac{1}{n}\int sinnxcosxdx=\{u=sinnx,\ dv=cosx\}=

    -cosxcosnx-\frac{1}{n}[sinnxsinx-\frac{1}{n}\int cosnxcosx]=.......$

    Could someone show me how to compute this integral?
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  2. #2
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    Re: inegral with 2 sine fcns

    $\displaystyle \sin x \cos nx = 1/2\,\sin \left( x+nx \right) -1/2\,\sin \left( -x+nx \right)$
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