The marginal revenue function of a commodity is `MR = 9 + 2x - 6x^(2)`, find the total revenue function.

Answer

To find the total revenue function from the given marginal revenue function \( MR = 9 + 2x - 6x^2 \), we will integrate the marginal revenue function with respect to \( x \).

### Step-by-Step Solution:

1. **Identify the Marginal Revenue Function**:
\[ MR = 9 + 2x - 6x^2 \]

2. **Set Up the Integral for Total Revenue**:
Since marginal revenue is the derivative of total revenue, we have:
\[ TR = \int MR \, dx = \int (9 + 2x - 6x^2) \, dx \]

3. **Integrate Each Term**:
We will integrate each term separately:
- The integral of \( 9 \) with respect to \( x \):
\[ \int 9 \, dx = 9x \]
- The integral of \( 2x \) with respect to \( x \):
\[ \int 2x \, dx = 2 \cdot \frac{x^2}{2} = x^2 \]
- The integral of \( -6x^2 \) with respect to \( x \):
\[ \int -6x^2 \, dx = -6 \cdot \frac{x^3}{3} = -2x^3 \]

4. **Combine the Results**:
Now, we combine the results of the integrals:
\[ TR = 9x + x^2 - 2x^3 + C \]
where \( C \) is the constant of integration.

5. **Final Total Revenue Function**:
Thus, the total revenue function is:
\[ TR = 9x + x^2 - 2x^3 + C \]