You should come up with the following substitution, such that:

`2x = t => 2dx = dt => dx = (dt)/2`

You need to change the limits of integration, such that:

`x = 0 => t = 0 `

`x = 2 => t = 4`

Changing the variable, yields:

`int_0^4 e^t*(dt)/2 = (1/2) int_0^4 e^t dt`

`(1/2) int_0^4 e^t dt = (1/2) e^t|_0^4`

Using the fundamental theorem of calculus, yields:

`(1/2) int_0^4 e^t dt = (1/2)(e^4 - e^0)`

`(1/2) int_0^4 e^t dt = (1/2)(e^4 - 1)`

**Hence, evaluating the given definite integral, yields` int_0^2 e^(2x) dx` yields **` int_0^2 e^(2x) dx = (1/2)(e^4 - 1).`

The definite integral is the area which has to be found, that is located between the given curve y = e^2x and the lines x = 0 and x = 2, also the x axis.

To calculate the area, we'll use the formula:

S = Integral (f(x) - ox)dx = Int f(x)dx = Int e^(2x) dx

Int e^(2x) dx = e^(2x)/2 + C

Now, we'll calculate the value of the area, using Leibnitz Newton formula::

S = F(2) - F(0), where

F(2) = e^(2*2)/2 = e^4/2

F(0) = e^(2*0)/2 = e^0/2 = 1/2

S = e^4/2 - 1/2

S = (e^4 - 1)/2

We have a difference of squares, at numerator:

S = (e^2-1)(e^2+1)/2

S = (e-1)(e+1)(e^2+1)/2