∫(x-2)√(x^2 4x 1) dx求不定积分,∫(x-2)√(x^2+4x+1)

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∫(x-2)√(x^2 4x 1) dx求不定积分,∫(x-2)√(x^2+4x+1)
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∫(x-2)√(x^2 4x 1) dx求不定积分,∫(x-2)√(x^2+4x+1)
∫(x-2)√(x^2 4x 1) dx
求不定积分,
∫(x-2)√(x^2+4x+1)

∫(x-2)√(x^2 4x 1) dx求不定积分,∫(x-2)√(x^2+4x+1)
∫(x-2)√(x^2+ 4x+ 1) dx
= (1/2)∫(2x+4)√(x^2+ 4x+ 1) dx - 4∫√(x^2+ 4x+ 1) dx
= (1/3)(x^2+ 4x+ 1)^(3/2) - 4∫√(x^2+ 4x+ 1) dx
consider
x^2+4x+1 = (x+2)^2 - 3
let
x+2 = √3secy
dx = √3secytany dy
∫√(x^2+ 4x+ 1) dx
=3∫(secy)^2 .tany dy
=3∫secy dsecy
=(3/2)(secy)^2 + C'
=(1/2)(x+2)^2 + C'
∫(x-2)√(x^2+ 4x+ 1) dx
= (1/3)(x^2+ 4x+ 1)^(3/2) - 4∫√(x^2+ 4x+ 1) dx
= (1/3)(x^2+ 4x+ 1)^(3/2) - 2(x+2)^2 + C

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