How do you evaluate ∫ e^(2x) dx?

To evaluate the integral ∫ e^(2x) dx, we apply the method of integration by substitution. Here, we notice that the function e^(2x) is an exponential function, and its derivative is directly related to itself through a chain rule.

First, consider the inner function of the exponent, which is 2x. We can set u = 2x. Therefore, the differential of u with respect to x is du/dx = 2, or dx = du/2. This substitution allows us to rewrite the integral.

Using this substitution, the integral becomes:

∫ e^(2x) dx = ∫ e^u (du/2) = (1/2) ∫ e^u du.

The integral of e^u with respect to u is e^u. Hence, we have:

(1/2) ∫ e^u du = (1/2)e^u + C.

Now, substituting back for u, where u = 2x, we get:

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

This shows that the correct evaluation of ∫ e^(2x) dx results in (1/2)e^(2x) +