题目

∫1/cos^3(t)dt

提问时间：2021-01-24

答案

∫1/cos^3(t)dt
= ∫sec^3(t)dt
= ∫sect dtant
= sect tant - ∫tant dsect
= sect tant - ∫tan^2 t sect dt
= sect tant - ∫(sec^2 t - 1) sect dt
= sect tant - ∫sec^3 t dt + ∫sect dt
= sect tant - ∫sec^3 t dt + ln|tant+sect|
所以
2∫sec^3 t dt = sect tant + ln|tant+sect|
∫sec^3 t dt = (1/2) sect tant + (1/2) ln|tant+sect|