\( \int \sec \theta d\theta\):
Note that \(d(\sec \theta) = (\sec \theta + \tan \theta) d\theta\) and \(d(\tan \theta) = \sec^2 \theta d\theta \)

Rewrite the expression by mutiplying \( \frac{\sec \theta + \tan \theta }{ \sec \theta +\tan \theta} = 1\)

\(\int \sec \theta d\theta = \int \sec \theta (\frac{\sec \theta + \tan \theta}{\sec \theta + \tan \theta}) d\theta\)

\(\int \sec \theta d\theta = \int \frac{\sec^2 \theta + \sec \theta \tan}{\sec \theta + \tan \theta} d\theta \)

Let \(u = \sec \theta + \tan \theta \)

\(du = \sec \theta \tan \theta d\theta +\sec^2 \theta d\theta \)

Thus \(\int \sec \theta d\theta = \ln{(\sec \theta + \tan \theta) +C}\)