Let's solve the given integral step by step.

Problem:

$\int 4x^3 (x^4 - 1)^2 dx$

Step 1: Substitution

Let: $u = x^4 - 1 \implies \frac{du}{dx} = 4x^3 \quad \text{or} \quad du = 4x^3 dx$

Substitute $u$ and $du$ into the integral: $\int 4x^3 (x^4 - 1)^2 dx = \int u^2 \, du$

Step 2: Solve the simpler integral

The integral of $u^2$ is: $\int u^2 \, du = \frac{u^3}{3} + C$

Step 3: Back-substitute $u = x^4 - 1$

$\frac{u^3}{3} + C = \frac{(x^4 - 1)^3}{3} + C$

Step 4: Multiply by the constant factor

The original integral includes the factor $4x^3$, which has already been accounted for in the substitution. Therefore, the final answer is: $\frac{(x^4 - 1)^3}{3} + C$

This matches the option: $\boxed{\frac{1}{3}(x^4 - 1)^3 + C}$

Let me know if you'd like further clarification or more details!

Relative Questions

1. What is the importance of substitution in solving integrals?
2. How do you identify the appropriate substitution in an integral?
3. Can this method be applied to similar polynomial integrals?
4. Why do we include the constant of integration ($C$)?
5. What happens if the limits of integration are provided?

Tip

Always check if substitution simplifies the integral into a standard form for easier calculation!