Answer
$$\tan \left( {{e^x} - 7} \right) + C $$
Work Step by Step
$$\eqalign{
& \int {{e^x}{{\sec }^2}\left( {{e^x} - 7} \right)} dx \cr
& {\text{integrate by substitution method}} \cr
& {\text{set }}u = {e^x} - 7{\text{ then }}\frac{{du}}{{dx}} = {e^x},\,\,\,\,dx = \frac{{du}}{{{e^x}}} \cr
& {\text{write the integrand in terms of }}u \cr
& \int {{e^x}{{\sec }^2}\left( {{e^x} - 7} \right)} dx = \int {{e^x}{{\sec }^2}\left( u \right)} \left( {\frac{{du}}{{{e^x}}}} \right) \cr
& {\text{cancel common terms}} \cr
& = \int {{{\sec }^2}u} du \cr
& {\text{integrating}} \cr
& = \tan u + C \cr
& {\text{replace }}{e^x} - 7{\text{ for }}u \cr
& = \tan \left( {{e^x} - 7} \right) + C \cr} $$
