To find the integral of $\frac{x^2}{5} + x^3 \, dx$, integrate each term separately.

Solution:

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

1. For $\int \frac{x^2}{5} \, dx$: $\int \frac{x^2}{5} dx = \frac{1}{5} \int x^2 dx = \frac{1}{5} \cdot \frac{x^3}{3} = \frac{x^3}{15}.$

2. For $\int x^3 \, dx$: $\int x^3 dx = \frac{x^4}{4}.$

Now, combine the results:

$\int \left( \frac{x^2}{5} + x^3 \right) dx = \frac{x^3}{15} + \frac{x^4}{4} + C,$

where $C$ is the constant of integration.

Final Answer:

$\boxed{\frac{x^3}{15} + \frac{x^4}{4} + C}$

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Would you like a more detailed explanation or examples?

Related Questions:

1. What is the definite integral of $\frac{x^2}{5} + x^3$ over an interval, say $[0, 2]$?
2. How do you compute integrals involving fractions like $\frac{1}{x^2 + 1}$?
3. Can you explain the power rule for integration and its limitations?
4. What are the steps to check if an integration is done correctly?
5. How does substitution help simplify more complex integrals?

Tip:

When integrating polynomials, apply the power rule $\int x^n dx = \frac{x^{n+1}}{n+1} + C$, but ensure $n \neq -1$.