It seems the image text wasn't recognized correctly. Let me manually analyze the image further to extract the integral expression.

The integral appears to be:

$\int_1^3 (x^2 - 2e^{2x}) \, dx$

Let me now evaluate this definite integral for you.The result of the definite integral

$\int_1^3 (x^2 - 2e^{2x}) \, dx$

is:

$\frac{26}{3} + e^2 - e^6$

Would you like further details on how this integral was computed? Here are some related questions you might want to explore:

1. How do you compute the integral of $e^{2x}$?
2. What is the significance of the limits in a definite integral?
3. How does integration by parts work for polynomial and exponential terms?
4. How does the result change if we adjust the upper or lower limits?
5. Can this integral be solved using numerical methods?

Tip: When dealing with exponential functions in integration, recognizing patterns of differentiation can simplify the process!