∫1/cos^3(t)dt

用基本公式:
∫1/cos³t dt
=∫sec³tdt
=(1/2)(sint*sec²t)+(1/2)∫sectdt
=(1/2)(sint*1/cos²t)+(1/2)(ln|sect+tant|)+C
=(1/2)(sect*tant)+(1/2)ln|sect+tant|+C
=(1/2)(sect*tant+ln|sect+tant|)+C
详细就是:
∫1/cos³t dt
=∫sec³tdt
=∫(sect*sec²t)dt
=∫[sect(1+tan²t)]dt
=∫sectdt+∫sect*tan²tdt
=∫sectdt+∫tantd(sect)
=∫sectdt+sect*tant-∫sectd(tant)
=∫sectdt+sect*tant-∫sec³tdt
∵∫sec³tdt=∫sectdt+sect*tant-∫sec³tdt
∴2∫sec³tdt=∫sectdt+sect*tant
即∫sec³tdt=(1/2)(sect*tant+∫sectdt)
=(1/2)(sect*tant+ln|sect+tant|)+C

分子分母同乘cost，然后做代换x=sint，化成有理函数积分。

∫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|

∫1/cos^3(t)dt
=∫cos(t)/cos^4(t)dt
=∫cos(t)/(1-sin(t)^2)^2dt
=∫1/(1-sin(t)^2)^2d(sint)
=∫1/(1-2x^2+x^4)dx x=sin(t)
=∫1/{[1/(1-x)-1/(1+x)][1/(1-x)-1/(1+x)]}dx/4
={∫(1-x)^(-2)dx+∫(1+x)^(-2)dx-∫2/[(1+x)(1-x)]dx}/4
=[(1-x)^(-1)-(1+x)^(-1)-ln(1-x)+ln(1+x)]/4+C
==[(1-sin(t))^(-1)-(1+sin(t))^(-1)-ln(1-sin(t))+ln(1+sin(t))]/4+C