To integrate the expression \( 2x(4x - 7) \, dx \), let's follow these steps:

### Step 1: Expand the expression
First, expand the product inside the integral:

\[
2x(4x - 7) = 2x \cdot 4x - 2x \cdot 7 = 8x^2 - 14x
\]

### Step 2: Integrate term by term
Now, integrate each term separately:

\[
\int (8x^2 - 14x) \, dx = \int 8x^2 \, dx - \int 14x \, dx
\]

- For \( \int 8x^2 \, dx \):
  \[
  \int 8x^2 \, dx = 8 \cdot \frac{x^3}{3} = \frac{8x^3}{3}
  \]

- For \( \int 14x \, dx \):
  \[
  \int 14x \, dx = 14 \cdot \frac{x^2}{2} = 7x^2
  \]

### Step 3: Combine the results
Putting it all together:

\[
\int 2x(4x - 7) \, dx = \frac{8x^3}{3} - 7x^2 + C
\]

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

### Final Answer:
\[
\int 2x(4x - 7) \, dx = \frac{8x^3}{3} - 7x^2 + C
\]

Would you like more details, or do you have any questions?

Here are 5 related questions:
1. How do you integrate a polynomial function in general?
2. What is the geometric interpretation of the integral?
3. How would the integration change if there were additional constants inside the expression?
4. Can you find the definite integral of this function over a specific interval?
5. How does the constant of integration affect the solution?

**Tip:** Always expand expressions before integrating if it simplifies the integration process.

Learn how to integrate the expression 2x(4x - 7) dx with a detailed step-by-step solution using integration by parts and the power rule. This problem covers polynomial integration techniques suitable for advanced high school students.

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