x=sect,则
∫dx/[x+√(x^2-1)]
=∫sect*tantdt/(sect+tant)
=∫sect*tant(sect-tant)dt
=∫[(sect)^2tant-sect(tant)^2]dt
=∫tantd(tant)-∫(sect)^3dt+∫sectdt
=(1/2)(tant)^2+ln│sect+tant│-∫(sect)^3dt
计算∫(sect)^3dt
=∫sectd(tant)
=secttant-∫(tant)^2*sectdt
=secttant-∫(sect)^3dt+∫sectdt
=secttant-∫(sect)^3dt+ln│sect+tant│
所以∫(sect)^3dt=（1/2)secttant+(1/2)ln│sect+tant│
代入得∫dx/[x+√(x^2-1)]
=(1/2)tan^2t-(1/2)secttant+(1/2)ln│sect+tant│+C
=(1/2)x^2-(1/2)x√(x^2-1)+(1/2)ln│x+√(x^2-1)│+C1(C1=C-1/2)
