Find the total revenue function for the marginal revenue function given by `M R = 20 e^(-(x)/(10))(1-(x)/(10))`.

Answer

To find the total revenue function \( R(x) \) from the given marginal revenue function \( MR(x) = 20 e^{-\frac{x}{10}} \left(1 - \frac{x}{10}\right) \), we need to integrate the marginal revenue function.

### Step-by-Step Solution:

1. **Write down the marginal revenue function**:
\[ MR(x) = 20 e^{-\frac{x}{10}} \left(1 - \frac{x}{10}\right) \]

2. **Set up the integral for total revenue**:
The total revenue function \( R(x) \) is the integral of the marginal revenue function:
\[ R(x) = \int MR(x) \, dx = \int 20 e^{-\frac{x}{10}} \left(1 - \frac{x}{10}\right) \, dx \]

3. **Distribute the terms inside the integral**:
\[ R(x) = \int 20 e^{-\frac{x}{10}} \, dx - \int 2x e^{-\frac{x}{10}} \, dx \]

4. **Integrate the first term**:
To integrate \( \int 20 e^{-\frac{x}{10}} \, dx \), we can use the substitution \( u = -\frac{x}{10} \), which gives \( du = -\frac{1}{10} dx \) or \( dx = -10 du \):
\[ \int 20 e^{-\frac{x}{10}} \, dx = -200 e^{-\frac{x}{10}} + C_1 \]

5. **Integrate the second term**:
For \( \int 2x e^{-\frac{x}{10}} \, dx \), we can use integration by parts:
Let \( u = x \) and \( dv = 2 e^{-\frac{x}{10}} dx \).
Then \( du = dx \) and \( v = -20 e^{-\frac{x}{10}} \).
Applying integration by parts:
\[ \int 2x e^{-\frac{x}{10}} \, dx = -20x e^{-\frac{x}{10}} + \int 20 e^{-\frac{x}{10}} \, dx \]
We already calculated \( \int 20 e^{-\frac{x}{10}} \, dx \):
\[ = -20x e^{-\frac{x}{10}} - 200 e^{-\frac{x}{10}} + C_2 \]

6. **Combine the results**:
Now, substituting back into the total revenue function:
\[ R(x) = -200 e^{-\frac{x}{10}} + 20x e^{-\frac{x}{10}} + 200 e^{-\frac{x}{10}} + C \]
The constants will cancel out, leaving:
\[ R(x) = 20x e^{-\frac{x}{10}} + C \]

7. **Determine the constant of integration \( C \)**:
To find \( C \), we can use the condition that when \( x = 0 \), \( R(0) = 0 \):
\[ R(0) = 20(0)e^{0} + C = 0 \implies C = 0 \]

8. **Final total revenue function**:
Thus, the total revenue function is:
\[ R(x) = 20x e^{-\frac{x}{10}} \]