∫sec³xdx求解,

Question

∫sec³xdx求解,

Answer 1

∫sec³xdx
=∫secxdtanx
=secxtanx-∫tanxdsecx
=secxtanx-∫tan²xsecxdx
=secxtanx-∫(sec²x-1)secxdx
=secxtanx-∫sec³xdx+∫secxdx
=> ∫sec³xdx=1/2*secxtanx+1/2∫secxdx
∫secxdx=∫1/cosx*dx
=∫1/cos²x*dsinx
=∫1/(1-sin²x)*dsinx
=1/2∫[1/(1-sinx)+1/(1+sinx)]*dsinx
=1/2{-∫[1/(1-sinx)]d(1-sinx)+∫[1/(1+sinx)]*d(1+sinx)}
=1/2{-ln(1-sinx)+ln(1+sinx)}+C
=1/2ln[(1+sinx)/(1-sinx)]+C
=> ∫sec³xdx=1/2*secxtanx+1/2∫secxdx
=1/2*secxtanx+1/4ln[(1+sinx)/(1-sinx)]+C