The problem asks us to identify which of the given integral formulas best resembles the integral $3 \int 2^{(3x+1)} dx$.

1. Problem Description

The problem asks us to identify which of the given integral formulas best resembles the integral $3 \int 2^{(3x+1)} dx$.

2. Solution Steps

We are given the integral $3 \int 2^{(3x+1)} dx$. We need to see which integral formula fits this form.

Let's examine each of the given options:

A. $\int u \, dv = uv - \int v \, du$ (Integration by parts) - This formula is used for integrating the product of two functions, which is not the case here.

B. $\int u^n \, du = \frac{u^{n+1}}{n+1} + C, n \neq -1$ (Power rule) - This formula is for integrating a power of a function. We have an exponential function.

C. $\int e^u \, du = e^u + C$ (Integral of exponential function with base e) - This is similar to our integral, but the base is 2, not $e$.

D. $\int a^u \, du = \frac{a^u}{\ln a} + C$ (Integral of exponential function with base a) - This formula looks like it fits our integral, where $a$ is a constant.

In our integral, $3 \int 2^{(3x+1)} dx$, we can see that the base is $a=2$ and $u = 3x+1$. Thus, the formula $\int a^u \, du = \frac{a^u}{\ln a} + C$ is the one that best resembles our given integral.

3. Final Answer

D. $\int a^u \, du = \frac{a^u}{\ln a} + C$