The integral you're asking to evaluate is:

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

To solve this, let's apply a substitution method.

Step-by-Step Solution:

1. Substitution: Let $u = x^3 + 2$. Then: $du = 3x^2 \, dx \quad \Rightarrow \quad \frac{du}{3} = x^2 \, dx$

2. Rewrite the integral: Using this substitution, the integral becomes: [ \int \frac{x^2}{\sqrt[4]{x^3 + 2}} , dx = \int \frac{1}{\sqrt[4]{u}} \cdot \frac{du}{3} ] which simplifies to: $\frac{1}{3} \int u^{-\frac{1}{4}} \, du$

3. Solve the integral: The integral of $u^{-\frac{1}{4}}$ is: $\frac{1}{3} \cdot \left( \frac{u^{\frac{3}{4}}}{\frac{3}{4}} \right) = \frac{4}{9} u^{\frac{3}{4}}$

4. Substitute back $u = x^3 + 2$: $\frac{4}{9} (x^3 + 2)^{\frac{3}{4}} + C$

Final Answer:

$\int \frac{x^2}{\sqrt[4]{x^3 + 2}} \, dx = \frac{4}{9} (x^3 + 2)^{\frac{3}{4}} + C$

Would you like further clarification on any steps or additional details?

Here are 5 related questions:

1. How can substitution be chosen effectively for complex integrals?
2. What are other methods for solving integrals besides substitution?
3. How would you approach an integral if the substitution method didn’t work?
4. Can you explain the rules for integrating powers of $u$?
5. What happens if the function inside the integral is more complicated, like trigonometric functions?

Tip: Always check the derivative after substitution to ensure it matches the original function!