What is the result of ∫ e^(3x) dx?

To find the integral ∫ e^(3x) dx, we can apply the technique of substitution or recognize the pattern of the integral.

The function e^(3x) is an exponential function, and when differentiating e^(kx) with respect to x, we get ke^(kx). Therefore, when integrating, we should account for the constant factor in the exponent.

To perform the integration correctly, we can use substitution. Let u = 3x, which implies that du = 3dx or dx = (1/3) du. As a result, the integral transforms as follows:

∫ e^(3x) dx = ∫ e^u * (1/3) du = (1/3) ∫ e^u du.

The integral of e^u with respect to u is simply e^u + C, so we have:

(1/3)(e^u + C) = (1/3)e^(3x) + C.

The constant of integration, C, remains because indefinite integrals include it. Thus, the result is (1/3)e^(3x) + C, which corresponds to the correct answer.

This method confirms that the right answer correctly applies the