请大家帮我看看这个定积分怎么做:在(0,π)区间证明 sin(2n+1)x/sinx的积分=π,
人气:330 ℃ 时间:2020-01-30 23:21:20
解答
和差化积公式
sin(2n+1)x=sinx-sinx+sin3x-sin3x+sin5x-sin5x+sin7x-sin7x+...+sin(2n-1)x-sin(2n-1)x+sin(2n+1)x
=sinx+cos2xsinx+cos4xsinx+cos6xsinx+.+cos2nxsinx
∫[0—〉π]{[sin(2n+1)x]/sinx}dx
=∫[0—〉π][1+cos2x+cos4x+cos6x+.+cos2nx]
=π
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