The integral provided in the image is:

$\int 7x^2 \cdot 4\sqrt[4]{12 + 4x^3} \, dx$

Let's break it down to solve:

Step 1: Simplify the expression

The given integral involves a product of two functions: $7x^2$ and $4\sqrt[4]{12 + 4x^3}$. The term $\sqrt[4]{12 + 4x^3}$ can be written as $(12 + 4x^3)^{1/4}$.

Thus, the integral becomes:

$\int 7x^2 \cdot 4 (12 + 4x^3)^{1/4} \, dx$

Simplify the constant:

$28 \int x^2 (12 + 4x^3)^{1/4} \, dx$

Step 2: Use substitution

Let $u = 12 + 4x^3$. Then, $du = 12x^2 \, dx$, and $x^2 \, dx = \frac{du}{12}$.

Substituting into the integral:

$28 \int (12 + 4x^3)^{1/4} \cdot \frac{du}{12}$

This simplifies to:

$\frac{28}{12} \int u^{1/4} \, du$

Simplifying the constant:

$\frac{7}{3} \int u^{1/4} \, du$

Step 3: Integrate

Now, integrate $u^{1/4}$:

$\frac{7}{3} \cdot \frac{u^{5/4}}{5/4} + C = \frac{7}{3} \cdot \frac{4}{5} u^{5/4} + C = \frac{28}{15} u^{5/4} + C$

Step 4: Substitute $u$ back

Recall that $u = 12 + 4x^3$. Substituting back:

$\frac{28}{15} (12 + 4x^3)^{5/4} + C$

Final Answer:

The correct answer is:

$\boxed{C \, \frac{28}{15} (12 + 4x^3)^{5/4} + C}$

Do you want more details on any step or have any questions?

Here are some related questions you might find useful:

1. How do you apply substitution for complex integrals?
2. What is the general rule for integrating expressions with fractional exponents?
3. How do you handle constants when simplifying integrals?
4. Can substitution always simplify integrals with polynomial functions?
5. How do you verify an integral result using differentiation?

Tip: Always simplify constants as much as possible before integrating to reduce computational complexity.