What is the result of the integral ∫ 3e^(2x) dx?

To solve the integral ∫ 3e^(2x) dx, we can use the technique of integration by substitution.

First, let’s recognize that the integral involves the exponential function e^(2x). When it comes to integrating functions of the form e^(ax), where a is a constant, the result generally follows the rule:

∫ e^(ax) dx = (1/a)e^(ax) + C.

In this case, we have the function multiplied by a constant, specifically 3 in this integral. Thus, we can factor the constant out of the integral:

∫ 3e^(2x) dx = 3 ∫ e^(2x) dx.

Now applying the integration rule stated above, we have:

∫ e^(2x) dx = (1/2)e^(2x) + C.

Substituting this back into the equation gives:

3 ∫ e^(2x) dx = 3 * (1/2)e^(2x) + C = (3/2)e^(2x) + C.

Therefore, the correct result of the integral ∫ 3e^(2x) dx is indeed (3/2)e^(2x) +