Answer
$$\sin \theta - \cos \theta + C$$
Work Step by Step
$$\eqalign{
& \int {\frac{{1 + \tan \theta }}{{\sec \theta }}} d\theta \cr
& {\text{split the numerator}} \cr
& = \int {\left( {\frac{1}{{\sec \theta }} + \frac{{\tan \theta }}{{\sec \theta }}} \right)} d\theta \cr
& {\text{use trigonometric identities sec}}\theta = \frac{1}{{\cos \theta }},{\text{ tan}}\theta = \frac{{\sin \theta }}{{\cos \theta }} \cr
& = \int {\left( {\frac{1}{{1/\cos \theta }} + \frac{{\sin \theta /\cos \theta }}{{1/\cos \theta }}} \right)} d\theta \cr
& = \int {\left( {\cos \theta + \sin \theta } \right)} d\theta \cr
& = \int {\cos \theta } d\theta + \int {\sin \theta } d\theta \cr
& {\text{fint the antiderivatives}} \cr
& = \sin \theta - \cos \theta + C \cr} $$
