The integral in the uploaded image is:

$\int x^2 \cdot \sqrt[3]{1 + x^3} \, dx$

Solution:

We'll solve this integral by using substitution:

1. Substitute: Let $u = 1 + x^3$.
   Then, $du = 3x^2 \, dx$, or equivalently $x^2 \, dx = \frac{du}{3}$.

2. Rewrite the integral: Substituting $u$ and $du$ into the integral: [ \int x^2 \cdot \sqrt[3]{1 + x^3} , dx = \int \sqrt[3]{u} \cdot \frac{du}{3}. ]

   Simplify the constant: $\frac{1}{3} \int u^{1/3} \, du.$

3. Integrate: The antiderivative of $u^{1/3}$ is: $\int u^{1/3} \, du = \frac{u^{4/3}}{4/3} = \frac{3}{4} u^{4/3}.$

   Therefore: $\frac{1}{3} \int u^{1/3} \, du = \frac{1}{3} \cdot \frac{3}{4} u^{4/3} = \frac{1}{4} u^{4/3}.$

4. Back-substitute $u = 1 + x^3$: $\frac{1}{4} u^{4/3} = \frac{1}{4} (1 + x^3)^{4/3}.$

5. Final answer: $\int x^2 \cdot \sqrt[3]{1 + x^3} \, dx = \frac{1}{4} (1 + x^3)^{4/3} + C,$ where $C$ is the constant of integration.

Do you want a detailed explanation of any of these steps?

Related Questions:

1. What are the conditions for applying substitution in an integral?
2. How do you handle integrals with cube roots or other fractional exponents?
3. Can this integral be solved using any other techniques (e.g., by parts)?
4. What is the geometric interpretation of the solution of this integral?
5. How does this technique generalize for integrals of the form $x^n \cdot \sqrt[m]{1 + x^k}$?

Tip:

When using substitution, always check that the differential $du$ matches all terms in the integral to avoid mistakes.