Answer
$$e^{\sin x}+c$$
Work Step by Step
Given
$$\int \frac{e^{\sin x}}{\sec x }dx $$
Let $u= \sin x \ \ \Rightarrow \ \ du=\cos x dx \ \Rightarrow dx=\sec x du$ ,then
\begin{align*}
\int \frac{e^{\sin x}}{\sec x }dx &=\int e^udu\\
&=e^u+c\\
&=e^{\sin x}+c
\end{align*}
