To solve the integral

$\int (x^3 + 6x)^5 (6x + 12) \, dx,$

let us proceed step by step.

Step 1: Substitution

Let: $u = x^3 + 6x.$ Then, the derivative of $u$ is: $\frac{du}{dx} = 3x^2 + 6 \quad \text{or} \quad du = (3x^2 + 6) \, dx.$

Notice that $6x + 12 = 6(x + 2)$, and: $3x^2 + 6 = 6x + 12.$

Thus: $du = (6x + 12) \, dx.$

Step 2: Rewrite the Integral

Substituting $u$ and $du$ into the integral: $\int (x^3 + 6x)^5 (6x + 12) \, dx = \int u^5 \, du.$

Step 3: Solve the Integral

The integral of $u^5$ is: $\int u^5 \, du = \frac{u^6}{6} + C.$

Step 4: Back-Substitute $u$

Substituting back $u = x^3 + 6x$, we get: $\frac{(x^3 + 6x)^6}{6} + C.$

Final Answer:

$\boxed{\frac{(x^3 + 6x)^6}{6} + C}$

Let me know if you would like further clarification! 😊

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Questions for Deeper Understanding:

1. Why does substitution simplify the integral?
2. How do you confirm that the derivative matches the original integral's components?
3. What are some tips for identifying substitution candidates in integrals?
4. How does the power rule apply to solve $\int u^5 \, du$?
5. What happens if we make an incorrect substitution? How can it be identified?

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Tip:

When faced with a composite function in an integral, look for its derivative or a multiple of it to use substitution effectively.