To solve the integral

$\int_{0}^{2} e^{2x} \cos(x - 2) \, dx$

we'll use integration techniques for functions of the form $e^{ax} \cos(bx + c)$. This generally involves integration by parts or leveraging known integration formulas for such expressions.

The solution process will be carried out as follows:

1. Rewrite the integral using known identities if needed.
2. Apply integration techniques like integration by parts or look up a formula for integrating $e^{ax} \cos(bx + c)$.
3. Evaluate the result from 0 to 2.

Let's calculate this and round it to two decimal places.