sin^4x(e^x/1+e^x)的定积分,上限是π/2,下限是-π/2
来源:学生作业帮 编辑:拍题作业网作业帮 分类:数学作业 时间:2024/04/19 19:15:56
sin^4x(e^x/1+e^x)的定积分,上限是π/2,下限是-π/2
I = ∫(-π/2->π/2) (e^x)sin⁴x/(1+e^x) dx
Let x = -y,dx = -dy
=> ∫(π/2->-π/2) (e^-y)sin⁴(-y) / (1+e^-y) (-dy)
= ∫(-π/2->π/2) sin⁴y/(1+e^y) dy
= ∫(-π/2->π/2) sin⁴x/(1+e^x) dx = J
I = (1/2)(I+J)
= (1/2)[∫(-π/2->π/2) (e^x)sin⁴x/(1+e^x) dx + ∫(-π/2->π/2) sin⁴x/(1+e^x) dx]
= (1/2)∫(-π/2->π/2) (1+e^x)sin⁴x/(1+e^x) dx
= (1/2)∫(-π/2->π/2) sin⁴x dx
= (2)(1/2)∫(0->π/2) [(1-cos2x)/2)]² dx
= (1/4)∫(0->π/2) (1-2cos2x+cos²2x) dx
= (1/4)∫(0->π/2) (1-2cos2x) dx + (1/8)∫(0->π/2) (1+cos4x) dx
= (1/4)(x - sin2x) + (1/8)(x + 1/4 * sin4x)
= (1/4)(π/2 - sinπ) + (1/8)(π/2 + 1/4 * sin2π)
= π/8 + π/16
= 3π/16
Let x = -y,dx = -dy
=> ∫(π/2->-π/2) (e^-y)sin⁴(-y) / (1+e^-y) (-dy)
= ∫(-π/2->π/2) sin⁴y/(1+e^y) dy
= ∫(-π/2->π/2) sin⁴x/(1+e^x) dx = J
I = (1/2)(I+J)
= (1/2)[∫(-π/2->π/2) (e^x)sin⁴x/(1+e^x) dx + ∫(-π/2->π/2) sin⁴x/(1+e^x) dx]
= (1/2)∫(-π/2->π/2) (1+e^x)sin⁴x/(1+e^x) dx
= (1/2)∫(-π/2->π/2) sin⁴x dx
= (2)(1/2)∫(0->π/2) [(1-cos2x)/2)]² dx
= (1/4)∫(0->π/2) (1-2cos2x+cos²2x) dx
= (1/4)∫(0->π/2) (1-2cos2x) dx + (1/8)∫(0->π/2) (1+cos4x) dx
= (1/4)(x - sin2x) + (1/8)(x + 1/4 * sin4x)
= (1/4)(π/2 - sinπ) + (1/8)(π/2 + 1/4 * sin2π)
= π/8 + π/16
= 3π/16
sin^4x(e^x/1+e^x)的定积分,上限是π/2,下限是-π/2
积分x/(sinx)^2上限π/3,下限π/4 定积分e^x/2 / 根号下1+e^(-x)上限1,下限0 定积分(1-
∫sinx/(1+e^(-x))dx 定积分 下限是-π/4,上限是π/4
定积分∫(e^(x²))(2x²+1)上限是0下限是-1
求定积分:ʃ |sin(x+1)|dx,上限是2π,下限是0.
设fx的一个原函数是Ln^2 X,求定积分xf'(x)dx 上限e下限1
定积分 绝对值sin x 上限 2π 下限 0
∫(e^(-x^2))dx上限是1,下限是0的积分是多少
定积分(上限e^2,下限e)lnx/(x-1)^2dx
高中定积分的计算计算1/(x-1)的定积分,上限e+1下限2,要过程
计算定积分 ∫ x ln(1+e^x) dx (上限2下限-2)
求(根号x+1/x)dx的定积分,上限是2,下限是1