To solve the integral

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

we will first expand the integrand and then integrate term by term.

Step 1: Expand the integrand

We use the distributive property to multiply $(x^3 + 2x)$ by $(3x^2 + 2)$.

$(x^3 + 2x)(3x^2 + 2) = x^3(3x^2 + 2) + 2x(3x^2 + 2)$

Now, distribute each term:

$x^3(3x^2) + x^3(2) + 2x(3x^2) + 2x(2)$

This gives:

$3x^5 + 2x^3 + 6x^3 + 4x$

Combine like terms:

$3x^5 + 8x^3 + 4x$

Step 2: Integrate term by term

Now, integrate each term individually:

$\int 3x^5 \, dx = \frac{3x^6}{6} = \frac{x^6}{2}$

$\int 8x^3 \, dx = \frac{8x^4}{4} = 2x^4$

$\int 4x \, dx = \frac{4x^2}{2} = 2x^2$

Step 3: Combine the results

Now, put everything together:

$\int (x^3 + 2x)(3x^2 + 2) \, dx = \frac{x^6}{2} + 2x^4 + 2x^2 + C$

where $C$ is the constant of integration.

Final Answer:

$\boxed{\frac{x^6}{2} + 2x^4 + 2x^2 + C}$

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Would you like a further breakdown of the steps or have any questions on the solution?

Here are some related questions:

1. How do you integrate polynomial expressions?
2. What is the power rule for integration?
3. How do you handle definite integrals with similar forms?
4. What happens if we have a higher degree term, like $x^7$?
5. Can you explain how to apply integration by parts in some cases?

Tip: When dealing with integrals of products of polynomials, it's often easiest to expand and simplify before applying basic integration rules.