Evaluate the integral of e^2x dx.

The integral of e^2x dx is (1/2)e^2x + C, where C is the constant of integration.

To evaluate the integral of e^2x dx, we can use the power rule of integration, which states that the integral of x^n dx is (1/(n+1))x^(n+1) + C. In this case, we have e^2x, which can be written as (e^2)^x. Using the power rule, we can integrate e^2x as follows:

∫ e^2x dx = (1/2) ∫ (e^2)^x d(2x)

Let u = e^2x and du/dx = 2e^2x. Then, d(2x) = 2dx and dx = (1/2)d(2x). Substituting these values, we get:

(1/2) ∫ u d(2x) = (1/2) ∫ u du/dx d(2x)

= (1/2) ∫ u d(u)

= (1/2) (u^2/2) + C

= (1/2)(e^2x)^2/2 + C

= (1/2)e^4x/4 + C

= (1/8)e^4x + C

Therefore, the integral of e^2x dx is (1/2)e^2x + C, where C is the constant of integration.