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b)Calcula la solución analítica de la integral impropia e indica si converge o diverge.  Anexar el procedimiento completo. 

The function given in the image is:

\[
f(x) = \frac{1}{x^2 + 1}
\]

We are asked to find the analytical solution of an improper integral involving this function and determine if it converges or diverges. 

Let's assume the improper integral is:

\[
I = \int_{-\infty}^{\infty} \frac{1}{x^2 + 1} \, dx
\]

### Step-by-Step Solution:

1. **Recognize the standard form of the function:**

   The integrand \(\frac{1}{x^2 + 1}\) is a standard function whose integral is known. The antiderivative of \(\frac{1}{x^2 + 1}\) is \(\arctan(x) + C\), where \(C\) is the constant of integration.

2. **Set up the improper integral:**

   Since this is an improper integral over the entire real line, we can express it as:

   \[
   I = \lim_{a \to -\infty} \lim_{b \to \infty} \int_{a}^{b} \frac{1}{x^2 + 1} \, dx
   \]

3. **Find the antiderivative:**

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

4. **Evaluate the definite integral:**

   Now apply the limits from \(a\) to \(b\):

   \[
   I = \lim_{a \to -\infty} \lim_{b \to \infty} \left[ \arctan(x) \right]_{a}^{b}
   \]

   This becomes:

   \[
   I = \lim_{a \to -\infty} \lim_{b \to \infty} \left( \arctan(b) - \arctan(a) \right)
   \]

5. **Evaluate the limits:**

   We know that:

   \[
   \lim_{x \to \infty} \arctan(x) = \frac{\pi}{2}, \quad \text{and} \quad \lim_{x \to -\infty} \arctan(x) = -\frac{\pi}{2}
   \]

   Substituting these limits, we get:

   \[
   I = \left( \frac{\pi}{2} - \left( -\frac{\pi}{2} \right) \right) = \pi
   \]

6. **Determine convergence or divergence:**

   The improper integral converges to \(\pi\).

### Conclusion:

The solution to the improper integral is:

\[
\int_{-\infty}^{\infty} \frac{1}{x^2 + 1} \, dx = \pi
\]

This integral converges, and its value is \(\pi\).

Would you like more details or have any other questions?

### Additional Questions:

1. What are the properties of the function \(f(x) = \frac{1}{x^2 + 1}\) that make the integral converge?
2. How does the convergence behavior of this integral change if the function is modified, e.g., \(f(x) = \frac{1}{x^2}\)?
3. Can you compute other improper integrals involving similar functions?
4. How does the convergence of this integral relate to its graphical behavior?
5. What would be the result of integrating \(f(x) = \frac{1}{x^2 + 1}\) over a finite interval instead?

### Tip:

When working with improper integrals, always check if the limits exist and evaluate carefully to determine if the integral converges or diverges.

Explore the analytical solution of the improper integral \(\int_{-\infty}^{\infty} \frac{1}{x^2 + 1} \, dx\) and determine if it converges or diverges. Detailed step-by-step solution with limits and antiderivative evaluation. Suitable for advanced undergraduate students and anyone interested in calculus and improper integrals.

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