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Math Help - computing integrals with sinx and sin(nx)

  1. #1
    Senior Member MacstersUndead's Avatar
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    computing integrals with sinx and sin(nx)

    For a Fourier method problem, I need to solve the following integral for the closed forms of coefficients c_n

    \int_0^\pi \! sin^2(x)sin(nx) \, \mathrm{d} x.

    For n even, the integral is 0 since sin(nx) would be odd about \pi/2
    For n odd, I let n = 2m + 1 and hence my integral with the sin(a+b) identity gives

    \int_0^\pi \! sin^2(x)sin(2mx)cos(x) + sin^2(x)cos(2mx)sin(x) \, \mathrm{d} x.

    but here I'm not sure what to do next. perhaps there is an easier way to approach this problem. any suggestions would be greatly appreciated.
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  2. #2
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    Re: computing integrals with sinx and sin(nx)

    You could try integrating by parts with:

    u = Sin x
    v' = Sin x * Sin nx = 0.5[Cos (n-1)x - Cos (n+1)x]

    I haven't tried it, but solving for this particular v is a common task in fourier problems.
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  3. #3
    Senior Member MacstersUndead's Avatar
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    Re: computing integrals with sinx and sin(nx)

    I shall try this. thank you very much.
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