To solve the given integral using the substitution $t = 3x + 1$, let's go step by step.

Problem:

$\int 12(3x+1)^3 \, dx$

Step 1: Substitution

Let $t = 3x + 1$. Then: $dt = 3 \, dx \quad \text{or} \quad dx = \frac{dt}{3}.$

Step 2: Rewrite the integral

Substitute $t = 3x + 1$ and $dx = \frac{dt}{3}$: $\int 12(3x+1)^3 \, dx = \int 12t^3 \cdot \frac{dt}{3}.$

Simplify the constants: $\int 12t^3 \cdot \frac{1}{3} \, dt = \int 4t^3 \, dt.$

Step 3: Integrate

The integral of $4t^3$ is: $\int 4t^3 \, dt = 4 \cdot \frac{t^4}{4} = t^4.$

So: $\int 4t^3 \, dt = t^4 + C.$

Step 4: Back-substitute $t = 3x + 1$

Finally, replace $t$ with $3x + 1$: $t^4 = (3x+1)^4.$

Thus, the solution is: $\int 12(3x+1)^3 \, dx = (3x+1)^4 + C.$

Would you like me to expand or clarify any part of the explanation? Let me know!

Related Questions:

1. What is the importance of substitution in integration problems?
2. How do you determine the correct substitution for a given integral?
3. Can this integral be solved using a different method, like expanding $(3x+1)^3$?
4. How do constants outside the integral affect the integration process?
5. What are common mistakes to avoid during substitution in integration?

Tip:

Always verify your solution by differentiating the result to see if you get back the original integrand.