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To solve the problem, we need to first differentiate the given function \((x^2 - x)^3\), and then use that result to evaluate the integral \(\int (2x - 1)(x^2 - x)^2 \, dx\).

### Step 1: Differentiate \((x^2 - x)^3\)

Let's denote \(u = x^2 - x\). The given function is \(u^3\).

To differentiate \((x^2 - x)^3\), we will use the chain rule:

\[
\frac{d}{dx} \left( (x^2 - x)^3 \right) = 3(x^2 - x)^2 \cdot \frac{d}{dx} (x^2 - x)
\]

Now, let's find \(\frac{d}{dx} (x^2 - x)\):

\[
\frac{d}{dx} (x^2 - x) = 2x - 1
\]

Substitute this back into the derivative:

\[
\frac{d}{dx} \left( (x^2 - x)^3 \right) = 3(x^2 - x)^2 \cdot (2x - 1)
\]

### Step 2: Use the Result to Evaluate the Integral

We are asked to find the integral:

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

Notice that the expression \((2x - 1)(x^2 - x)^2\) matches exactly with the derivative we just found:

\[
\frac{d}{dx} \left( (x^2 - x)^3 \right) = 3(x^2 - x)^2 (2x - 1)
\]

Thus:

\[
(2x - 1)(x^2 - x)^2 \, dx = \frac{1}{3} \, d \left( (x^2 - x)^3 \right)
\]

Now, the integral simplifies to:

\[
\int (2x - 1)(x^2 - x)^2 \, dx = \frac{1}{3} \int d \left( (x^2 - x)^3 \right)
\]

This is just:

\[
\frac{1}{3} (x^2 - x)^3 + C
\]

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

### Final Answer

1. The derivative of \((x^2 - x)^3\) is:

   \[
   \frac{d}{dx} \left( (x^2 - x)^3 \right) = 3(x^2 - x)^2 (2x - 1)
   \]

2. The integral \(\int (2x - 1)(x^2 - x)^2 \, dx\) is:

   \[
   \frac{1}{3} (x^2 - x)^3 + C
   \]

Would you like more details or have any questions?

Here are some related questions to explore:

1. How do you apply the chain rule to different composite functions?
2. What are other common integrals that use the chain rule in reverse?
3. Can you explain more examples involving polynomial differentiation and integration?
4. How does substitution work in integrals?
5. What are the different techniques for solving integrals involving polynomials?

**Tip:** When differentiating composite functions, remember to differentiate the outer function first and multiply by the derivative of the inner function.

Differentiate (x^2 - x)^3, and hence find ∫(2x - 1)(x^2 - x)^2 dx.

This problem involves differentiating the function (x^2 - x)^3 using the chain rule, followed by evaluating the integral ∫(2x - 1)(x^2 - x)^2 dx. Step-by-step solutions include the use of key differentiation and integration concepts.

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