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The given problem asks to find the value of the integral \( I = \int_0^1 \frac{1}{1 + x^2} \, dx \) using Gaussian quadrature for \( n = 2 \) and \( n = 4 \).

### Steps:
1. **Gaussian Quadrature Formula:**
   The Gaussian quadrature approximates an integral as:
   \[
   \int_{-1}^1 f(x) \, dx \approx \sum_{i=1}^n w_i f(x_i)
   \]
   where \( w_i \) are the weights and \( x_i \) are the roots of the Legendre polynomials \( P_n(x) \) over the interval \([-1, 1]\).

2. **Transforming the Interval:**
   The given integral is over \([0, 1]\), not \([-1, 1]\). We apply a change of variables to transform the limits:
   \[
   x = \frac{1+t}{2}, \quad dx = \frac{1}{2} dt
   \]
   Then the integral becomes:
   \[
   I = \frac{1}{2} \int_{-1}^1 \frac{1}{1 + \left(\frac{1+t}{2}\right)^2} \, dt
   \]

3. **Applying Gaussian Quadrature:**
   For \( n = 2 \) and \( n = 4 \), we use the weights and points for Gaussian quadrature from standard tables. After substituting, we evaluate the integral approximation for each case.

Let me compute the values for \( n = 2 \) and \( n = 4 \).The values of the integral \( I = \int_0^1 \frac{1}{1+x^2} \, dx \) using Gaussian quadrature are:

- For \( n = 2 \): \( I \approx 0.7869 \)
- For \( n = 4 \): \( I \approx 0.7854 \)

These are approximations of the integral based on different numbers of quadrature points. The value converges closer to the true integral as \( n \) increases.

Do you have any questions or would you like a detailed explanation of any part of the process?

Here are five related questions for further exploration:
1. How does Gaussian quadrature compare to other numerical integration methods like Simpson's rule?
2. What is the exact value of \( \int_0^1 \frac{1}{1+x^2} \, dx \)?
3. How do the weights and points in Gaussian quadrature change for higher values of \( n \)?
4. Can Gaussian quadrature be applied to multidimensional integrals?
5. How do we derive the Legendre polynomials used in Gaussian quadrature?

**Tip:** As \( n \) increases in Gaussian quadrature, the approximation becomes more accurate, but the computation cost also rises due to the higher number of function evaluations.

Find the value of the integral I = ∫[0, 1] 1 / (1 + x^2) dx using Gauss quadrature for n = 2 and n = 4.

Learn how to approximate the integral I = ∫[0, 1] 1 / (1 + x^2) dx using Gauss quadrature with n=2 and n=4. Explore the Gaussian Quadrature method, Legendre polynomials, and transformations used in numerical integration. This solution is suitable for college-level calculus and numerical methods.

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