Answer
$e^{\sin x}+C$
Work Step by Step
$\displaystyle \int\frac{e^{\sin x}}{\sec x}dx=\int e^{\sin x}\cdot\cos dx=\left[\begin{array}{l}
t=\sin x\\
dt=\cos xdx
\end{array}\right]$
$=\displaystyle \int e^{t}dt$
$=e^{t}+C$
$=e^{\sin x}+C$