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.