We are asked to evaluate the integral:

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

Step 1: Simplify the integrand using substitution

Let's apply substitution to make the integral more manageable. Set:

$u = 4 + 2x^4$

Now differentiate $u$ with respect to $x$:

$\frac{du}{dx} = 8x^3$

Thus, we can rewrite $dx$ in terms of $du$:

$dx = \frac{du}{8x^3}$

Now, substitute $u$ and $du$ into the integral:

$\int 2x^3 (u)^3 \cdot \frac{du}{8x^3}$

Step 2: Cancel terms

The $x^3$ terms cancel out, and we can factor out constants:

$\int \frac{2}{8} u^3 \, du = \frac{1}{4} \int u^3 \, du$

Step 3: Integrate

The integral of $u^3$ is straightforward:

$\frac{1}{4} \cdot \frac{u^4}{4} = \frac{u^4}{16}$

Step 4: Substitute $u$ back

Recall that $u = 4 + 2x^4$, so substitute back into the result:

$\frac{(4 + 2x^4)^4}{16} + C$

Thus, the integral of $2x^3(4 + 2x^4)^3 \, dx$ is:

$\boxed{\frac{(4 + 2x^4)^4}{16} + C}$

Would you like any more details or clarifications?

Here are 5 related questions for further exploration:

1. How do you solve integrals using the substitution method in general?
2. What is the chain rule in differentiation and how does it relate to integration?
3. Can integration by parts be used for this problem? Why or why not?
4. How would you approach integrating trigonometric functions using substitution?
5. How does integration relate to finding the area under a curve?

Tip: When using substitution in integration, make sure to express both the function and the differential completely in terms of the new variable before solving.