Answer
$${e^{\tan x}} + C $$
Work Step by Step
$$\eqalign{
& \int {{{\sec }^2}} \left( x \right){e^{\tan x}}dx \cr
& {\text{integrate by the substitution method}} \cr
& {\text{set }}u = \tan x{\text{ then }}\frac{{du}}{{dx}} = {\sec ^2}x,\,\,\,\,dx = \frac{{du}}{{{{\sec }^2}x}} \cr
& {\text{write the integrand in terms of }}u \cr
& \int {{{\sec }^2}} \left( x \right){e^{\tan x}}dx = \int {{{\sec }^2}} \left( x \right){e^u}\left( {\frac{{du}}{{{{\sec }^2}x}}} \right) \cr
& {\text{cancel common terms}} \cr
& = \int {{e^u}} du \cr
& {\text{integrating }} \cr
& = {e^u} + C \cr
& {\text{replace }}\tan x{\text{ for }}u \cr
& = {e^{\tan x}} + C \cr} $$
