兩題都給你做下：
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∫ xsin(nx) dx
= (- 1/n)∫ x d[cos(nx)]
= (- x/n)cos(nx) + (1/n)∫ cos(nx) dx
= (- x/n)cos(nx) + (1/n^2)sin(nx) + C
∴
∫(- π→0) xsin(nx) dx
= (- x/n)cos(nx) + (1/n^2)sin(nx)_(x = 0) - (- x/n)cos(nx) + (1/n^2)sin(nx)_(x = - π)
= (- π/n)cos(nπ)
= - π/n,當(dāng)n為偶數(shù)
= π/n,當(dāng)n為奇數(shù)
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∫ xcos(nx) dx
= (1/n)∫ x d[sin(nx)]
= (x/n)sin(nx) - (1/n)∫ sin(nx) dx
= (x/n)sin(nx) + (1/n^2)cos(nx) + C
∴
∫(- π→0) xcos(nx) dx
= (x/n)sin(nx) + (1/n^2)cos(nx)_(x = 0) - (x/n)sin(nx) + (1/n^2)cos(nx)_(x = - π)
= (1/n^2)[1 - cos(nπ)]
= 0,當(dāng)n為偶數(shù)
= 2/n^2,當(dāng)n為奇數(shù)