To solve the given indefinite integral:

$\int x^2 \left(x^3 + 5\right)^{3/2} dx$

Solution:

1. Substitution: Let $u = x^3 + 5$. Then, $du = 3x^2 dx$, or $x^2 dx = \frac{1}{3} du$.

2. Rewriting the integral: Substitute $u$ and $x^2 dx$: $\int x^2 \left(x^3 + 5\right)^{3/2} dx = \int \left(u^{3/2}\right) \frac{1}{3} du$

   Simplify: $\frac{1}{3} \int u^{3/2} du$

3. Integrate: Use the power rule for integration: $\int u^n du = \frac{u^{n+1}}{n+1}, \quad \text{for } n \neq -1$

   Here, $n = \frac{3}{2}$, so: $\int u^{3/2} du = \frac{u^{5/2}}{5/2} = \frac{2}{5} u^{5/2}$

   Multiply by $\frac{1}{3}$: $\frac{1}{3} \cdot \frac{2}{5} u^{5/2} = \frac{2}{15} u^{5/2}$

4. Substitute back: Recall $u = x^3 + 5$: $\frac{2}{15} (x^3 + 5)^{5/2} + C$

Final Answer:

$\int x^2 \left(x^3 + 5\right)^{3/2} dx = \frac{2}{15} (x^3 + 5)^{5/2} + C$

Would you like further clarification or a step-by-step breakdown of any part?

Related Questions:

1. How can substitution simplify other types of integrals?
2. What are the conditions for applying the power rule in integration?
3. How do we identify the best substitution for a given integral?
4. Can you provide examples where substitution doesn't work as effectively?
5. How do definite integrals differ from indefinite ones when solving using substitution?

Tip:

When using substitution, always ensure that the derivative of your substitution matches part of the integrand for smoother integration!