Answer
$$\frac{1}{\pi }\sin \pi x + C$$
Work Step by Step
$$\eqalign{
& \int {\frac{1}{{\sec \pi x}}} dx \cr
& {\text{trigonometric identity }}\sec \theta = \frac{1}{{\cos \theta }} \cr
& = \int {\cos \pi x} dx \cr
& {\text{substitute }}u = \pi x,{\text{ }}du = \pi dx \cr
& = \int {\cos } u\left( {\frac{1}{\pi }dx} \right) \cr
& = \frac{1}{\pi }\int {\cos } udu \cr
& {\text{find antiderivative}} \cr
& = \frac{1}{\pi }\sin u + C \cr
& {\text{write in terms of }}x \cr
& = \frac{1}{\pi }\sin \pi x + C \cr} $$
