The integral given is:

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

We are told the answer is:

$-\frac{1}{9} (3x + 2)^{-3} + C$

Let's explain why this is correct step-by-step.

Step 1: Recognizing the general form

The given expression fits the general integral form of:

$\int u^n \, du = \frac{u^{n+1}}{n+1} + C$

Where $u = 3x + 2$ and $n = -4$. However, in this case, we must account for the chain rule when integrating.

Step 2: Substitution (u-substitution)

Let: $u = 3x + 2$

Now, differentiate $u$ with respect to $x$: $\frac{du}{dx} = 3$

Thus, $dx = \frac{du}{3}$.

Step 3: Substitute into the integral

Now substitute into the integral:

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

Factor out the constant $\frac{1}{3}$:

$\frac{1}{3} \int u^{-4} \, du$

Step 4: Apply the power rule of integration

Using the power rule for integrals, where $\int u^n \, du = \frac{u^{n+1}}{n+1} + C$, apply this to $u^{-4}$:

$\frac{1}{3} \cdot \frac{u^{-3}}{-3} + C = \frac{1}{3} \cdot \left( -\frac{1}{3} u^{-3} \right) + C$

This simplifies to:

$-\frac{1}{9} u^{-3} + C$

Step 5: Substitute $u = 3x + 2$ back

Now substitute $u = 3x + 2$ back into the equation:

$-\frac{1}{9} (3x + 2)^{-3} + C$

This matches the given answer.

Conclusion

The answer is correct because we applied the substitution method, used the power rule for integrals, and accounted for the chain rule by adjusting for the derivative of $u = 3x + 2$.

Would you like further details or clarifications?

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Here are five questions to further explore this topic:

1. How does the chain rule apply when solving integrals involving substitution?
2. What happens if you forget to account for the derivative of the inner function during substitution?
3. How does the constant $\frac{1}{3}$ affect the integral process?
4. What are the steps to solve similar integrals involving negative exponents?
5. Why is the constant of integration $C$ always necessary in indefinite integrals?

Tip: Always check if substitution is needed when you see a composite function inside an integral.