Chapter 4 - Integration - 4.2 The Indefinite Integral - Exercises Set 4.2 - Page 279: 25

$$\tan \theta + C$$

$$\eqalign{ & \int {\frac{{\sec \theta }}{{\cos \theta }}} d\theta \cr & = \int {\frac{1}{{\cos \theta }}\sec \theta } d\theta \cr & {\text{basic trigonometric identity sec}}\theta = \frac{1}{{\cos \theta }} \cr & = \int {\frac{1}{{\cos \theta }}\frac{1}{{\cos \theta }}} d\theta \cr & = \int {\frac{1}{{{{\cos }^2}\theta }}} d\theta \cr & = \int {{{\sec }^2}\theta } d\theta \cr & {\text{find the antiderivative}} \cr & = \tan \theta + C \cr} $$