To find the total revenue, $R(x)$, given the marginal revenue $MR = \frac{-48}{(2x + 2)^2} + 48$, we need to integrate the marginal revenue function with respect to $x$.

1. Set up the integral: $R(x) = \int MR \, dx = \int \left( \frac{-48}{(2x + 2)^2} + 48 \right) dx$

2. Integrate each term separately:

   • For the term $\frac{-48}{(2x + 2)^2}$: Let $u = 2x + 2$, so $du = 2 \, dx$ or $dx = \frac{du}{2}$. This transforms the integral to: $\int \frac{-48}{(2x + 2)^2} \, dx = \int \frac{-48}{u^2} \cdot \frac{1}{2} \, du = -24 \int \frac{1}{u^2} \, du = -24 \cdot \left( \frac{-1}{u} \right) = \frac{24}{2x + 2}$

   • For the constant term $48$: $\int 48 \, dx = 48x$

3. Combine the results: $R(x) = \frac{24}{2x + 2} + 48x + C$

4. Simplify the expression: $R(x) = \frac{24}{2(x + 1)} + 48x + C$

So, the total revenue function $R(x)$ is: $R(x) = \frac{24}{2x + 2} + 48x + C$ where $C$ is the constant of integration.

Would you like me to delve deeper into finding $C$ based on initial conditions or explore further?

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1. What would the value of $C$ be if we know the initial revenue?
2. How would the total revenue change with respect to $x$?
3. Can we interpret the behavior of $MR$ as $x$ increases?
4. How can this marginal revenue formula help optimize profit?
5. What does the concavity of $R(x)$ tell us about revenue growth?

Tip: Always check the limits if calculating definite integrals for real applications.