\(\text{The given integral is,}\)

\(I=\int\frac{e^{\sin x}}{\sec x}dx\)

\(\text{Using trigonometric identity, }\cos\theta=\frac{1}{\sec\theta}\)

\(I=\int\cos x(e^{\sin x})dx\)

\(\text{Using substitution method,}\)

\(e^{\sin x}=t\)

\(\frac{d}{dx}(e^{\sin x})=\frac{dt}{dx}\)

Let\([\frac{d}{d(\sin x)}(e^{\sin x})][\frac{d}{dx}(\sin x)]=\frac{dt}{dx}\)

\((e^{\sin x})(\cos x)=\frac{dt}{dx}\)

\((e^{\sin x})(\cos x)dx=dt\)

\(\text{Then the given integral gets transformed as, }\)

\(I=\int dt=t+C\)

\(\text{Putting back the value of t, we get}\)

\(I=e^{\sin x}+C\)

\(\text{Therefore, the value of the given integral is }e^{\sin x}+C\)

\(I=\int\frac{e^{\sin x}}{\sec x}dx\)

\(\text{Using trigonometric identity, }\cos\theta=\frac{1}{\sec\theta}\)

\(I=\int\cos x(e^{\sin x})dx\)

\(\text{Using substitution method,}\)

\(e^{\sin x}=t\)

\(\frac{d}{dx}(e^{\sin x})=\frac{dt}{dx}\)

Let\([\frac{d}{d(\sin x)}(e^{\sin x})][\frac{d}{dx}(\sin x)]=\frac{dt}{dx}\)

\((e^{\sin x})(\cos x)=\frac{dt}{dx}\)

\((e^{\sin x})(\cos x)dx=dt\)

\(\text{Then the given integral gets transformed as, }\)

\(I=\int dt=t+C\)

\(\text{Putting back the value of t, we get}\)

\(I=e^{\sin x}+C\)

\(\text{Therefore, the value of the given integral is }e^{\sin x}+C\)