Step 1

The given integral can be evaluated by the method of substitution.

We know that,

\(\displaystyle\int{e}^{{{x}}}{\left.{d}{x}\right.}={e}^{{{x}}}+{c}\)

\(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\sin{{x}}}={\cos{{x}}}\)

Step 2

The given integral is,

\(\displaystyle{I}=\int{\cos{{x}}}{e}^{{{\sin{{x}}}}}{\left.{d}{x}\right.}\).

Put,

\(\displaystyle{\sin{{x}}}={u}\)

\(\displaystyle{d}{u}={\cos{{x}}}{\left.{d}{x}\right.}\)

Then,

\(\displaystyle{I}=\int{e}^{{{u}}}{d}{u}\)

\(\displaystyle={e}^{{{u}}}+{c}\)

\(\displaystyle={e}^{{{\sin{{x}}}}}+{c}\)

Hence, the required expression is, \(\displaystyle{e}^{{{\sin{{x}}}}}+{c}\).

The given integral can be evaluated by the method of substitution.

We know that,

\(\displaystyle\int{e}^{{{x}}}{\left.{d}{x}\right.}={e}^{{{x}}}+{c}\)

\(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\sin{{x}}}={\cos{{x}}}\)

Step 2

The given integral is,

\(\displaystyle{I}=\int{\cos{{x}}}{e}^{{{\sin{{x}}}}}{\left.{d}{x}\right.}\).

Put,

\(\displaystyle{\sin{{x}}}={u}\)

\(\displaystyle{d}{u}={\cos{{x}}}{\left.{d}{x}\right.}\)

Then,

\(\displaystyle{I}=\int{e}^{{{u}}}{d}{u}\)

\(\displaystyle={e}^{{{u}}}+{c}\)

\(\displaystyle={e}^{{{\sin{{x}}}}}+{c}\)

Hence, the required expression is, \(\displaystyle{e}^{{{\sin{{x}}}}}+{c}\).