∫（0到-π）x sin（nx）dx＝?

题目

∫（0到-π）x sin（nx）dx＝?
如上,把sin（nx）改为cos（nx）=?5

答案

两题都给你做下：
————————————————————————————————————
∫ 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,当n为偶数
= π/n,当n为奇数
————————————————————————————————————
∫ 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,当n为偶数
= 2/n^2,当n为奇数