The standard approach for showing $\int \sec \theta \, \mathrm d \theta = \ln|\sec \theta + \tan \theta| + C$ is to multiply by $\dfrac{\sec \theta + \tan \theta}{\sec \theta + \tan \theta}$ and then do a substitution with $u = \sec \theta + \tan \theta$.
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 $\int \csc \theta \,\mathrm d \theta$. Does anyone know of another way to evaluate $\int \sec \theta \, \mathrm d \theta$?
Best Answer
Another way is:
$$\int \sec x \,dx = \int \frac{\cos x}{\cos^2 x} \,dx = \int \frac{\cos x}{1-\sin^2 x} \,dx = \frac{1}{2} \int \left( \frac{1}{1-\sin x} + \frac{1}{1+\sin x} \right) \cos x dx $$ $$= \frac{1}{2} \log \left| \frac{1+\sin x}{1-\sin x} \right| + C.$$
It's worth noting that the answer can appear in many disguises. Another is $$\log \left| \tan \left( \frac{\pi}{4} + \frac{x}{2} \right) \right| $$