It looks like you want to integrate the function \(8x^3 + 12x^2\). To clarify, we are solving the following indefinite integral:

\[
\int (8x^3 + 12x^2) \, dx
\]

### Step-by-Step Solution:

1. **Break down the integral:**

   The integral can be separated into two simpler integrals:

   \[
   \int (8x^3 + 12x^2) \, dx = \int 8x^3 \, dx + \int 12x^2 \, dx
   \]

2. **Integrate each term separately:**

   - For \(\int 8x^3 \, dx\), we use the power rule of integration:

     \[
     \int x^n \, dx = \frac{x^{n+1}}{n+1} + C
     \]

     Applying this rule:

     \[
     \int 8x^3 \, dx = 8 \cdot \frac{x^{3+1}}{3+1} = 8 \cdot \frac{x^4}{4} = 2x^4
     \]

   - For \(\int 12x^2 \, dx\), applying the power rule again:

     \[
     \int 12x^2 \, dx = 12 \cdot \frac{x^{2+1}}{2+1} = 12 \cdot \frac{x^3}{3} = 4x^3
     \]

3. **Combine the results:**

   Now, we sum the integrated terms:

   \[
   \int (8x^3 + 12x^2) \, dx = 2x^4 + 4x^3 + C
   \]

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

### Final Answer:

\[
\int (8x^3 + 12x^2) \, dx = 2x^4 + 4x^3 + C
\]

Would you like more details or have any questions?

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Here are 5 related questions you might be interested in:

1. How do you integrate functions with multiple terms, like trinomials or higher-degree polynomials?
2. What are the integration rules for exponential and logarithmic functions?
3. How do definite integrals differ from indefinite integrals?
4. What techniques can be used to integrate functions that are products of simpler functions (e.g., integration by parts)?
5. How do you apply substitution to solve integrals that aren't straightforward?

**Tip:** When integrating, always remember to add the constant of integration \(C\) for indefinite integrals.

Learn how to integrate the polynomial 8x^3 + 12x^2 step-by-step. This problem involves using the power rule of integration and is suitable for undergraduate students studying calculus.

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