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∫根号（cosx-（cosx）^3）dx上限π/2下限-π/2
题目详情
∫根号（cosx-（cosx）^3） dx 上限π/2 下限-π/2
▼优质解答
答案和解析
∫ [π/2,-π/2] (cosx-cos^3x)^(1/2)dx
=∫ [π/2,-π/2](cosx*sin^2x)^(1/2)dx
=∫ [π/2,0](cosx)^(1/2)sinxdx+∫ [0,-π/2](cosx)^(1/2)(-sinx)dx
=-∫ [π/2,0](cosx)^(1/2)d(cosx)+∫ [0,-π/2](cosx)^(1/2)d(cosx)
= -2/3(cosx)^(3/2) [π/2,0] + 2/3(cosx)^(3/2)[0,-π/2]
=4/3
=∫ [π/2,-π/2](cosx*sin^2x)^(1/2)dx
=∫ [π/2,0](cosx)^(1/2)sinxdx+∫ [0,-π/2](cosx)^(1/2)(-sinx)dx
=-∫ [π/2,0](cosx)^(1/2)d(cosx)+∫ [0,-π/2](cosx)^(1/2)d(cosx)
= -2/3(cosx)^(3/2) [π/2,0] + 2/3(cosx)^(3/2)[0,-π/2]
=4/3
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