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The image shows the following problem:

**Problem Statement:**
Solve for \( x \) in the equation:
\[
\int (4x) \, dx = 26x^3 - 8
\]

### Solution:

1. **Interpret the Integral Expression:**
   The given equation involves an indefinite integral of \(4x\) with respect to \(x\):
   \[
   \int 4x \, dx
   \]

   To solve this, we first find the antiderivative of \(4x\).

2. **Compute the Antiderivative:**
   The integral of \(4x\) with respect to \(x\) is:
   \[
   \int 4x \, dx = 2x^2 + C
   \]
   where \(C\) is the constant of integration.

3. **Set the Antiderivative Equal to the Given Expression:**
   Now, we equate this to the expression on the right side of the equation:
   \[
   2x^2 + C = 26x^3 - 8
   \]

4. **Simplify and Solve for \(x\):**
   Rearrange the equation:
   \[
   2x^2 - 26x^3 + C + 8 = 0
   \]

   This is a cubic equation in terms of \(x\). However, since we have an arbitrary constant \(C\), we cannot find a specific numeric solution for \(x\) without additional information about \(C\).

Would you like to proceed with further steps, such as solving for \(x\) when a specific value for \(C\) is provided?

### Related Questions:
1. How do you find the general solution for a cubic equation?
2. What is the importance of the constant of integration when solving indefinite integrals?
3. How can you determine if a cubic equation has real roots?
4. How do you use numerical methods to solve cubic equations?
5. What are the applications of integrals in solving real-world problems?

**Tip:** When dealing with indefinite integrals, always remember to include the constant of integration \(C\).

Learn how to solve the integral equation involving a cubic expression and understand the concepts of antiderivatives and cubic equations. This problem requires advanced calculus knowledge and is suitable for Grades 11-12 students or equivalent.

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