To solve the integral:

$\int (x + 2)(x^2 - 1) \, dx$

Step 1: Expand the integrand

First, we distribute the terms inside the parentheses:

$(x + 2)(x^2 - 1) = x(x^2 - 1) + 2(x^2 - 1)$

Simplify each term:

$x(x^2 - 1) = x^3 - x$ $2(x^2 - 1) = 2x^2 - 2$

Now, combine all terms:

$x^3 - x + 2x^2 - 2$

So the integral becomes:

$\int (x^3 - x + 2x^2 - 2) \, dx$

Step 2: Integrate term by term

Now, integrate each term separately:

$\int x^3 \, dx = \frac{x^4}{4}$ $\int (-x) \, dx = -\frac{x^2}{2}$ $\int 2x^2 \, dx = \frac{2x^3}{3}$ $\int (-2) \, dx = -2x$

Step 3: Combine the results

Putting all the integrals together:

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

Where $C$ is the constant of integration.

Final answer:

$\int (x + 2)(x^2 - 1) \, dx = \frac{x^4}{4} - \frac{x^2}{2} + \frac{2x^3}{3} - 2x + C$

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Would you like to dive deeper into any of the steps? Here are some related questions:

1. How do you integrate polynomials in general?
2. What is the method of integration by parts, and how would it apply here?
3. How do you check if an integral is correct?
4. Can you explain the concept of the constant of integration?
5. What happens when we apply definite limits to this integral?

Tip: When expanding polynomials for integration, always combine like terms before integrating. This can simplify the process significantly!