Chapter 6 - Applications of Integration - 6.8 Logarithmic and Exponential - 6.8 Exercises - Page 480: 22

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

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