Ways to evaluate ∫secθdθ\int \sec \theta \, \mathrm d \theta

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?

Answer

Another way is:

secxdx=cosxcos2xdx=cosx1sin2xdx=12(11sinx+11+sinx)cosxdx
=12log|1+sinx1sinx|+C.

It’s worth noting that the answer can appear in many disguises. Another is
log|tan(π4+x2)|

Attribution
Source : Link , Question Author : Mike Spivey , Answer Author : KingLogic

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