To solve the integral:

\[
\int (2x - 1)^2 \, dx
\]

### Step 1: Expand the integrand
First, expand \((2x - 1)^2\):

\[
(2x - 1)^2 = (2x - 1)(2x - 1) = 4x^2 - 4x + 1
\]

So, the integral becomes:

\[
\int (4x^2 - 4x + 1) \, dx
\]

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

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

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

\[
\frac{4x^3}{3} - 2x^2 + x + C
\]

Thus, the solution to the integral is:

\[
\boxed{\frac{4x^3}{3} - 2x^2 + x + C}
\]

Do you want any details or have any questions?

Here are some related questions:
1. How do you expand binomials in general?
2. What are some other examples of polynomial integrals?
3. How would you solve an integral involving a cube instead of a square?
4. What is the geometric interpretation of an integral like this?
5. How do definite integrals differ when applied to this type of problem?

**Tip:** Always simplify or expand the integrand first to make integration easier.

Learn how to solve the integral of (2x-1)^2 with a detailed step-by-step solution. This problem covers polynomial expansion, integration techniques, and the use of the Fundamental Theorem of Calculus. Suitable for students in Grades 10-12.

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