急,求不定积分(1/X*根号(X^2-1))DX
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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
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
![急,求不定积分(1/X*根号(X^2-1))DX](/uploads/image/z/3218434-34-4.jpg?t=%E6%80%A5%2C%E6%B1%82%E4%B8%8D%E5%AE%9A%E7%A7%AF%E5%88%86%EF%BC%881%2FX%2A%E6%A0%B9%E5%8F%B7%EF%BC%88X%5E2-1%EF%BC%89%EF%BC%89DX)
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
∴∫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