作业帮求∫ (cot^5 xsin^4 x) dx.
来源：学生作业帮 编辑：作业帮 分类：数学作业 时间：2024/05/14 01:25:51
求∫ (cot^5 xsin^4 x) dx.
integral sin^4(x) cos^5(x) dx =
(3 sin(x))/128-1/192 sin(3 x)-1/320 sin(5 x)+(sin(7 x))/1792+(sin(9 x))/2304+C
再问： 是cot的五次方
再答： ∫cot^5(x).sin^4(x)dx = ∫[cos^5(x)/sin^5(x)]sin^4(x) dx = ∫cos^5(x)/sin(x) dx = ∫cos(x)cos^4(x)/sin(x) dx = ∫cos(x)(cos²(x))²/sin(x) dx = ∫cos(x)(1-sin²(x))²/sin(x) dx Let u=sin(x) du = cos(x)dx ∫cos(x)(1-sin²(x))²/sin(x) dx = ∫[(1-u²)²/u]du = ∫(1 - 2u² + u^4)/u du =∫1/u - 2u + u³ du = ln|u| - u² + (u^4)/4 + c = ln|sin(x)| - sin²(x) + sin^4(x)/4 + c
(3 sin(x))/128-1/192 sin(3 x)-1/320 sin(5 x)+(sin(7 x))/1792+(sin(9 x))/2304+C
再问： 是cot的五次方
再答： ∫cot^5(x).sin^4(x)dx = ∫[cos^5(x)/sin^5(x)]sin^4(x) dx = ∫cos^5(x)/sin(x) dx = ∫cos(x)cos^4(x)/sin(x) dx = ∫cos(x)(cos²(x))²/sin(x) dx = ∫cos(x)(1-sin²(x))²/sin(x) dx Let u=sin(x) du = cos(x)dx ∫cos(x)(1-sin²(x))²/sin(x) dx = ∫[(1-u²)²/u]du = ∫(1 - 2u² + u^4)/u du =∫1/u - 2u + u³ du = ln|u| - u² + (u^4)/4 + c = ln|sin(x)| - sin²(x) + sin^4(x)/4 + c
求∫ (cot^5 xsin^4 x) dx.
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