Answer
$$\frac{1}{2}\tan x + \frac{1}{2}x + C$$
Work Step by Step
$$\eqalign{
& \int {\frac{{\sec x + \cos x}}{{2\cos x}}} dx \cr
& {\text{distribute }} \cr
& = \int {\left( {\frac{{\sec x}}{{2\cos x}} + \frac{{\cos x}}{{2\cos x}}} \right)} dx \cr
& = \int {\left( {\frac{1}{{2\cos x}}\sec x + \frac{1}{2}} \right)} dx \cr
& {\text{basic trigonometric identity sec}}\theta = \frac{1}{{\cos \theta }} \cr
& = \int {\left( {\frac{1}{2}\sec x\sec x + \frac{1}{2}} \right)} dx \cr
& = \int {\left( {\frac{1}{2}{{\sec }^2}x + \frac{1}{2}} \right)} dx \cr
& = \int {\frac{1}{2}{{\sec }^2}x} dx + \int {\frac{1}{2}} dx \cr
& {\text{find the antiderivative}} \cr
& = \frac{1}{2}\tan x + \frac{1}{2}x + C \cr} $$
