急,求不定积分（1/X*根号（X^2-1））DX

题目

急,求不定积分（1/X*根号（X^2-1））DX
令x=sect
dx=sinx/(cosx)^2 dt
(x^2-1)=(sect)^2-1=(tanx)^2 根号(x^2-1)=tanx
Dx/(x*根号(x^2-1)) =dx/(sect*tant)=dx/(1/cost)*tanx=acecos(1/x)做出来了

答案

设t=1/x 则dx=-dt/t^2
∴∫1/[x(x^2-1)^(1/2)]dx
=-∫(dt/t^2)*t|t|/(1-t^2)
=-sgn(t)∫dt/(1-t^2)^(1/2)
=-sgn(x)arcsint+C
=-arcsin(1/|x|)+C