The standard approach for showing ∫secθdθ=ln|secθ+tanθ|+C is to multiply by secθ+tanθsecθ+tanθ and then do a substitution with u=secθ+tanθ.
I like the fact that this trick leads to a fast and clean derivation, but I also find it unsatisfying: It’s not very intuitive, nor does it seem to have applicability to any integration problem other than ∫cscθdθ. Does anyone know of another way to evaluate ∫secθdθ?
Another way is:
It’s worth noting that the answer can appear in many disguises. Another is