原式=∫(tanx)^2secxdx
=∫[(secx)^2-1]secxdx
=∫(secx)^3dx-∫secxdx
=∫(secx)^3dx-ln|tanx+secx|……①
而∫(secx)^3dx=∫secxd(tanx)
=secxtanx-∫tanxd(secx)
=secxtanx-∫[secx[(secx)^2-1]]dx
=secxtanx-∫(secx)^3dx+∫secxdx
=secxtanx+ln|tanx+secx|-∫(secx)^3dx
移项且两端除以2得
∫(secx)^3dx=(1/2)secxtanx+(1/2)ln|tanx+secx|+C…………②
将②式代回①式即为所求.