Chapter 7 - Principles Of Integral Evaluation - 7.1 An Overview Of Integration Methods - Exercises Set 7.1 - Page 490: 9

$${e^{\tan x}} + C$$

$$\eqalign{ & \int {{e^{\tan x}}{{\sec }^2}x} dx \cr & {\text{substitute }}u = \tan x,{\text{ }}du = {\sec ^2}xdx \cr & \int {{e^{\tan x}}{{\sec }^2}x} dx = \int {{e^u}du} \cr & {\text{find the antiderivative}} \cr & \int {{e^u}du} = {e^u} + C \cr & {\text{write in terms of }}x,{\text{ replace }}u = \tan x \cr & = {e^{\tan x}} + C \cr} $$