求 ∫√((x+1)/(x-1))+√((x-1)/(x+1))dx
2个回答

∫√{[(x+1)/(x-1)]+√[(x-1)/(x+1)]}dx

(x+1)/(x-1)＞0

x＞1或x＜-1

-1＜1/x＜1

设1/x=cosβ ,0＜β＜π

∫√{[(x+1)/(x-1)]+√[(x-1)/(x+1)]}dx

=∫√{[(1+1/x)/(1-1/x)]+√[(1-1/x)/(1+1/x)]}dx

=∫√{[(1+cosβ)/(1-cosβ)]+√[(1-cosβ)/(1+cosβ)]}d(1/cosβ)

=∫[cot(β/2)+tan(β/2)]d(1/cosβ)

=∫[cot(β/2)+tan(β/2)]d(1/cosβ)

=∫[cos(β/2)/sin(β/2)+sin(β/2)/cos(β/2)]d(1/cosβ)

=∫{1/[cos(β/2)sin(β/2)]}d(1/cosβ)

=2∫(1/sinβ)d(1/cosβ)

=-2∫[(1/sinβ)(cosβ)^(-2)]d(cosβ)

=2∫[(1/sinβ)(cosβ)^(-2)]sinβdβ

=2∫[(cosβ)^(-2)]dβ

=2tanβ+c

=2tanarccos(1/x)+c

=2sinarccos(1/x)/cosarccos(1/x)+c

=2xsinarccos(1/x)+c

=2x√{1-[cosarccos(1/x)]^2}+c

=2x√(1-1/x^2)+c

=2(x/|x|)√(x^2-1)+c

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