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Resulve los 3 problemas de Calculo Integral.

Let's solve the three integral problems step-by-step.

### 1. Integral 1

\[
\int \left(x^2 - \frac{1}{2}\right)^2 \, dx
\]

First, expand the integrand:

\[
\left(x^2 - \frac{1}{2}\right)^2 = x^4 - x^2 + \frac{1}{4}
\]

So, the integral becomes:

\[
\int \left(x^4 - x^2 + \frac{1}{4}\right) \, dx
\]

Integrating term by term:

\[
\int x^4 \, dx - \int x^2 \, dx + \int \frac{1}{4} \, dx = \frac{x^5}{5} - \frac{x^3}{3} + \frac{x}{4} + C
\]

where \( C \) is the constant of integration.

### Solution for Integral 1:
\[
\frac{x^5}{5} - \frac{x^3}{3} + \frac{x}{4} + C
\]

### 2. Integral 2

\[
\int \sqrt[3]{x^2} \left(x^2 + x + 1\right) \, dx
\]

First, rewrite the integrand:

\[
\sqrt[3]{x^2} = x^{\frac{2}{3}}
\]

Thus, the integral becomes:

\[
\int x^{\frac{2}{3}} (x^2 + x + 1) \, dx
\]

Distribute \( x^{\frac{2}{3}} \):

\[
\int \left(x^{\frac{8}{3}} + x^{\frac{5}{3}} + x^{\frac{2}{3}}\right) \, dx
\]

Integrate each term:

\[
\frac{x^{\frac{11}{3}}}{\frac{11}{3}} + \frac{x^{\frac{8}{3}}}{\frac{8}{3}} + \frac{x^{\frac{5}{3}}}{\frac{5}{3}} + C
\]

Simplify:

\[
\frac{3}{11}x^{\frac{11}{3}} + \frac{3}{8}x^{\frac{8}{3}} + \frac{3}{5}x^{\frac{5}{3}} + C
\]

### Solution for Integral 2:
\[
\frac{3}{11}x^{\frac{11}{3}} + \frac{3}{8}x^{\frac{8}{3}} + \frac{3}{5}x^{\frac{5}{3}} + C
\]

### 3. Integral 3

\[
\int \frac{\left(\frac{x}{2} + 1\right)^2}{x} \, dx
\]

Rewrite the integrand:

\[
\left(\frac{x}{2} + 1\right)^2 = \frac{x^2}{4} + x + 1
\]

So, the integral becomes:

\[
\int \frac{\frac{x^2}{4} + x + 1}{x} \, dx = \int \left(\frac{x}{4} + 1 + \frac{1}{x}\right) \, dx
\]

Integrate term by term:

\[
\frac{1}{4} \int x \, dx + \int 1 \, dx + \int \frac{1}{x} \, dx = \frac{1}{4} \frac{x^2}{2} + x + \ln|x| + C
\]

Simplify:

\[
\frac{x^2}{8} + x + \ln|x| + C
\]

### Solution for Integral 3:
\[
\frac{x^2}{8} + x + \ln|x| + C
\]

### Summary of Solutions:

1. \(\frac{x^5}{5} - \frac{x^3}{3} + \frac{x}{4} + C\)
2. \(\frac{3}{11}x^{\frac{11}{3}} + \frac{3}{8}x^{\frac{8}{3}} + \frac{3}{5}x^{\frac{5}{3}} + C\)
3. \(\frac{x^2}{8} + x + \ln|x| + C\)

Would you like more details or have any questions?

### Further Questions to Explore:

1. How would the integration change if the exponent of \(x\) was different in the integrals?
2. Can you derive the integrals using substitution methods instead?
3. What is the geometric interpretation of these integrals?
4. How do improper integrals work with these types of functions?
5. How can we apply integration techniques to solve real-world problems?

### Tip:
When dealing with integrals involving roots and powers, rewriting the expressions in terms of fractional exponents can often simplify the integration process.

Resuelve los 3 problemas de Cálculo Integral.

This guide provides a detailed solution to three integral calculus problems, covering algebraic expansion, fractional exponents, and polynomial integration. Ideal for undergraduate calculus students, the solutions demonstrate the application of power rules, logarithmic integration, and step-by-step explanations.

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