求广义积分∫(3,+∞)1/[(x-1)^4*√(x²-2x)]dx

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求广义积分∫(3,+∞)1/[(x-1)^4*√(x²-2x)]dx
求广义积分∫(3,+∞)1/[(x-1)^4*√(x²-2x)]dx

求广义积分∫(3,+∞)1/[(x-1)^4*√(x²-2x)]dx
∫dx/[(x-1)^4*√(x^2-2x)
=∫d(x-1)/[(x-1)^4*√((x-1)^2-1)]
(x-1)=secu sinu^2=1-1/(x-1)^2=(x^2-2x)/(x-1)^2 sinu=√(x^2-2x)/(x-1)=√[1-1/(x-1)^2]
=∫secutanudu/(secu^4*tanu)
=∫cosu^3du
=∫(1-sinu^2)dsinu
=sinu-(1/3)(sinu)^3
=√[1-1/(x-1)^2]-(1/3)√[1-1/(x-1)^2]^3
∫[3,+∝) dx/[(x-1)^4*√(x^2-x)]
=(1-1/3)-√3/2-(1/3)√(3/4)^3