Step-by-step Solution:

We are given the integral: \[ \int e^x \left( \frac{1 + \sin x \cos x}{\cos^2 x} \right) dx = \int e^x (\tan x + \sec^2 x) dx. \] We can use the result: \[ \int e^x [f(x) + f'(x)] \, dx = e^x f(x), \] where \( f(x) = \tan x \), and \( f'(x) = \sec^2 x \). Thus, the integral becomes: \[ \int e^x (\tan x + \sec^2 x) \, dx = e^x \tan x + C, \] where \( C \) is the constant of integration. Hence, the answer is: \[ e^x \tan x + C. \]