#### Answer

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

#### Work Step by Step

$$\eqalign{
& \int {\frac{{{e^{\sin x}}}}{{\sec x}}dx} \cr
& {\text{use trigonometric identities}} \cr
& = \int {{e^{\sin x}}\cos xdx} \cr
& {\text{substitute }}u = \sin x,{\text{ }}du = \cos xdx \cr
& = \int {{e^u}du} \cr
& {\text{find the antiderivative}} \cr
& = {e^u} + C \cr
& {\text{replace }}u = \sin x \cr
& = {e^{\sin x}} + C \cr} $$