If the Marginal revenue function for output x is given by 𝑀𝑅 = 
[4/(2𝑥+3)2
]−1, find the total revenue function and the demand 
function.

Given the marginal revenue function:

\[
MR = \left[\frac{4}{(2x + 3)^2}\right] - 1
\]

### 1. Find the Total Revenue (TR) Function

The total revenue function \( TR(x) \) is the integral of the marginal revenue function \( MR(x) \) with respect to \( x \).

\[
TR(x) = \int MR(x) \, dx
\]

Substituting \( MR(x) \):

\[
TR(x) = \int \left[\frac{4}{(2x + 3)^2} - 1\right] dx
\]

### 2. Solve the Integral

Let's solve the integral:

\[
TR(x) = \int \frac{4}{(2x + 3)^2} \, dx - \int 1 \, dx
\]

1. **First Integral**: \( \int \frac{4}{(2x + 3)^2} \, dx \)

   Let \( u = 2x + 3 \), so \( du = 2 \, dx \), or \( dx = \frac{du}{2} \).

   \[
   \int \frac{4}{(2x + 3)^2} \, dx = 4 \int \frac{1}{u^2} \cdot \frac{du}{2} = 2 \int \frac{1}{u^2} \, du
   \]

   The integral of \( \frac{1}{u^2} \) is \( -\frac{1}{u} \).

   \[
   2 \int \frac{1}{u^2} \, du = -\frac{2}{u} = -\frac{2}{2x + 3}
   \]

2. **Second Integral**: \( \int 1 \, dx = x \)

Thus, the total revenue function is:

\[
TR(x) = -\frac{2}{2x + 3} - x + C
\]

where \( C \) is the constant of integration.

### 3. Find the Demand Function

The demand function \( p(x) \) is the price function, which can be obtained by dividing the total revenue function \( TR(x) \) by the output \( x \).

\[
p(x) = \frac{TR(x)}{x}
\]

Substituting the expression for \( TR(x) \):

\[
p(x) = \frac{-\frac{2}{2x + 3} - x + C}{x}
\]

Simplify this to obtain the demand function:

\[
p(x) = -\frac{2}{x(2x + 3)} - 1 + \frac{C}{x}
\]

### Summary

- **Total Revenue Function**: 
\[
TR(x) = -\frac{2}{2x + 3} - x + C
\]

- **Demand Function**:
\[
p(x) = -\frac{2}{x(2x + 3)} - 1 + \frac{C}{x}
\]

Would you like further details or have any questions? 

Here are five related questions:

1. What does the constant \( C \) represent in the total revenue function?
2. How can you determine the constant \( C \) if given initial conditions?
3. What does the demand function tell us about the relationship between price and quantity?
4. How would the total revenue function change if the marginal revenue function were different?
5. Can the demand function be linear, and under what conditions?

**Tip**: Always check your integration results by differentiating to see if you retrieve the original function!

If the Marginal revenue function for output x is given by MR = [4/(2x+3)^2] − 1, find the total revenue function and the demand function.

Explore how to find the total revenue function and demand function from a given marginal revenue function MR = [4/(2x+3)^2] − 1. This problem involves key calculus concepts such as integration and revenue functions, suitable for college-level students.

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