大师作文网三道积分题!在线等!
来源：学生作业帮 编辑：大师作文网作业帮 分类：数学作业 时间：2024/11/13 18:43:01
三道积分题!在线等!
(30)
∫(0->π/2) [cost/√[1+(sint)^2] dt
let
sint = tany
cost dt = (secy)^2.dy
t=0,y=0
t=π/2,y=π/4
∫(0->π/2) [cost/√[1+(sint)^2] dt
=∫(0->π/4) secy dy
= [ln|secy + tany|] (0->π/4)
=ln(√2 + 1)
(16)
∫(√2/3->2/3) dx/[x^5.√(9x^2-1) ]
let
3x= secy
3dx= secy tany dy
x=√2/3 ,y =π/4
x= 2/3 ,y = π/3
∫(√2/3->2/3) dx/[x^5.√(9x^2-1) ]
=81∫(π/4->π/3) (cosy)^4 dy
=(81/4)∫(π/4->π/3) (1+cos2y)^2 dy
=(81/4)∫(π/4->π/3) [1+2cos2y+ (cos2y)^2 ]dy
=(81/8)∫(π/4->π/3) [3+4cos2y+cos4y ]dy
=(81/8) [3y+2sin2y+sin(4y)/ 4 ](π/4->π/3)
=(81/8) [ (π+√3 -√3/8) - (3π/4 +2) ]
=(81/8) [ π/4 +7√3/8 - 2]
=(81/64) [ 2π +7√3 - 16]
(8)
∫dt/[t^2√(t^2-16) ]
let
t= 4secy
dt = 4secytany dy
∫dt/[t^2√(t^2-16) ]
=(1/16)∫cosydy
=(1/16) siny + C
=(1/16) [√(t^2-16) / t] + C
∫(0->π/2) [cost/√[1+(sint)^2] dt
let
sint = tany
cost dt = (secy)^2.dy
t=0,y=0
t=π/2,y=π/4
∫(0->π/2) [cost/√[1+(sint)^2] dt
=∫(0->π/4) secy dy
= [ln|secy + tany|] (0->π/4)
=ln(√2 + 1)
(16)
∫(√2/3->2/3) dx/[x^5.√(9x^2-1) ]
let
3x= secy
3dx= secy tany dy
x=√2/3 ,y =π/4
x= 2/3 ,y = π/3
∫(√2/3->2/3) dx/[x^5.√(9x^2-1) ]
=81∫(π/4->π/3) (cosy)^4 dy
=(81/4)∫(π/4->π/3) (1+cos2y)^2 dy
=(81/4)∫(π/4->π/3) [1+2cos2y+ (cos2y)^2 ]dy
=(81/8)∫(π/4->π/3) [3+4cos2y+cos4y ]dy
=(81/8) [3y+2sin2y+sin(4y)/ 4 ](π/4->π/3)
=(81/8) [ (π+√3 -√3/8) - (3π/4 +2) ]
=(81/8) [ π/4 +7√3/8 - 2]
=(81/64) [ 2π +7√3 - 16]
(8)
∫dt/[t^2√(t^2-16) ]
let
t= 4secy
dt = 4secytany dy
∫dt/[t^2√(t^2-16) ]
=(1/16)∫cosydy
=(1/16) siny + C
=(1/16) [√(t^2-16) / t] + C