∫1/cos³xdx=∫1/cosxdtanx

=tanx/cosx-∫tanxd1/cosx

=tanx/cosx-∫tanx*sinx/cos²xdx

=tanx/cosx-∫sin²x/cos³xdx

=tanx/cosx-∫(1-cos²x)/cos³xdx

=tanx/cosx-∫1/cos³xdx+∫1/cosxdx

2∫1/cos³xdx=tanx/cosx+∫1/cosxdx

∫1/cosxdx=∫1/cos²xdsinx

=∫1/(1-sin²x)dsinx

=1/2∫1/(1+sinx)+1/(1-sinx)dsinx

=1/2[ln(1+sinx)+ln(1-sinx)]

所以2∫1/cos³xdx=tanx/cosx+∫1/cosxdx

=tanx/cosx+1/2[ln(1+sinx)+ln(1-sinx)]

∫1/cos³xdx=

tanx/2cosx+1/4[ln(1+sinx)+ln(1-sinx)]+C