Q3: Evaluate I = integral of dx/1+x^3 and limit is 3 to 1 using three-point Gauss Legendre rule.

To evaluate the integral \( I = \int_{1}^{3} \frac{dx}{1+x^3} \) using the three-point Gauss-Legendre rule, we proceed as follows:

### Step 1: Standardize the Integral
The Gauss-Legendre quadrature rule is generally applied over the interval \([-1, 1]\). To apply it over any interval \([a, b]\), we need to transform the integral.

Given \( a = 1 \) and \( b = 3 \), we use the transformation:
\[
x = \frac{b-a}{2} \xi + \frac{a+b}{2}
\]
where \( \xi \) varies from \(-1\) to \(1\).

The integral becomes:
\[
I = \frac{b-a}{2} \int_{-1}^{1} \frac{1}{1 + \left(\frac{b-a}{2} \xi + \frac{a+b}{2}\right)^3} d\xi
\]
Substitute \( a = 1 \) and \( b = 3 \):
\[
I = \frac{3-1}{2} \int_{-1}^{1} \frac{1}{1 + \left(\frac{2}{2} \xi + \frac{4}{2}\right)^3} d\xi
\]
Simplifying:
\[
I = \int_{-1}^{1} \frac{1}{1 + \left(\xi + 2\right)^3} d\xi
\]

### Step 2: Apply the Three-Point Gauss-Legendre Rule
The three-point Gauss-Legendre rule uses the following weights \(w_i\) and points \(x_i\):

\[
\begin{array}{c|c|c}
i & \xi_i & w_i \\
\hline
1 & -\sqrt{\frac{3}{5}} & \frac{5}{9} \\
2 & 0 & \frac{8}{9} \\
3 & \sqrt{\frac{3}{5}} & \frac{5}{9} \\
\end{array}
\]

Thus, the integral becomes:
\[
I \approx \sum_{i=1}^{3} w_i \cdot f\left(\xi_i\right)
\]
where \( f(\xi) = \frac{1}{1 + \left(\xi + 2\right)^3} \).

### Step 3: Compute the Approximation
Substitute the values for \(\xi_i\) and compute the function values:
\[
f(-\sqrt{3/5}) = \frac{1}{1 + \left(-\sqrt{3/5} + 2\right)^3}
\]
\[
f(0) = \frac{1}{1 + 2^3} = \frac{1}{9}
\]
\[
f(\sqrt{3/5}) = \frac{1}{1 + \left(\sqrt{3/5} + 2\right)^3}
\]

The integral approximation is:
\[
I \approx \frac{5}{9} \cdot f(-\sqrt{3/5}) + \frac{8}{9} \cdot f(0) + \frac{5}{9} \cdot f(\sqrt{3/5})
\]

### Step 4: Numerical Evaluation
Evaluate each term and sum them to get the final approximation.

Let me calculate this.The approximate value of the integral \( I = \int_{1}^{3} \frac{dx}{1+x^3} \) using the three-point Gauss-Legendre rule is approximately **0.3192**.

Would you like more details or have any questions?

Here are five related questions:
1. How does the three-point Gauss-Legendre rule compare with higher-point rules?
2. What are the advantages of using Gauss-Legendre quadrature over other numerical integration methods?
3. How does the choice of transformation affect the accuracy of the integral approximation?
4. Can Gauss-Legendre quadrature be extended to multidimensional integrals?
5. How would you apply the five-point Gauss-Legendre rule to this problem?

**Tip:** For more accurate results, using more points in Gauss-Legendre quadrature can significantly improve the approximation, especially for integrands with varying complexity.

Learn how to evaluate the integral using the three-point Gauss-Legendre rule with a detailed step-by-step solution. This problem covers numerical integration techniques and the application of Gauss-Legendre quadrature. Suitable for advanced undergraduate students and practitioners.

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