三道积分题!在线等!

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三道积分题!在线等!三道积分题!在线等!三道积分题!在线等!(30)∫(0->π/2)[cost/√[1+(sint)^2]dtletsint=tanycostdt=(secy)^2.dyt=0,y=

三道积分题!在线等!
三道积分题!在线等!

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

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