Answer
$$ - \cos {e^x} + C $$
Work Step by Step
$$\eqalign{
& \int {{e^x}\sin \left( {{e^x}} \right)} dx \cr
& {\text{integrate by the substitution method}} \cr
& {\text{set }}u = {e^x}{\text{ then }}\frac{{du}}{{dx}} = {e^x},\,\,\,\,dx = \frac{{du}}{{{e^x}}} \cr
& {\text{write the integrand in terms of }}u \cr
& \int {{e^x}\sin \left( {{e^x}} \right)} dx = \int {{e^x}\sin \left( u \right)} \left( {\frac{{du}}{{{e^x}}}} \right) \cr
& {\text{cancel common terms}} \cr
& = \int {{e^u}\left( {\frac{{du}}{\pi }} \right)} \cr
& = \int {\sin u} du \cr
& {\text{integrate}} \cr
& = - \cos u + C \cr
& {\text{replace }}{e^x}{\text{ for }}u \cr
& = - \cos {e^x} + C \cr} $$
