`int(3x+4)^(3)dx=`

Options

A. `((3x+4)^(4))/(12)+c`
B. `(3(3x+4)^(4))/(4)+c`
C. `(3(3x+4)^(2))/(2)+c`
D. `(3(3x+4)^(2))/(4)+c`

Answer

To solve the integral \( \int (3x + 4)^3 \, dx \), we can use the formula for integrating a function of the form \( (Ax + B)^n \).

### Step-by-Step Solution:

1. **Identify the components**:
We have \( A = 3 \), \( B = 4 \), and \( n = 3 \).

2. **Use the integration formula**:
The formula for integrating \( (Ax + B)^n \) is:
\[ \int (Ax + B)^n \, dx = \frac{(Ax + B)^{n+1}}{A(n+1)} + C \]
where \( C \) is the constant of integration.

3. **Apply the formula**:
Plugging in our values:
\[ \int (3x + 4)^3 \, dx = \frac{(3x + 4)^{3+1}}{3(3+1)} + C \]

4. **Simplify the expression**:
This becomes:
\[ \int (3x + 4)^3 \, dx = \frac{(3x + 4)^4}{3 \cdot 4} + C \]
\[ = \frac{(3x + 4)^4}{12} + C \]

5. **Final result**:
Thus, the integral evaluates to:
\[ \int (3x + 4)^3 \, dx = \frac{(3x + 4)^4}{12} + C \]