Find the integral I = fe- cos xdx using two-point Gauss Legendre rule. Marks

To find the integral \( I = \int e^{-\cos x} \, dx \) using the two-point Gauss-Legendre rule, we can follow these steps:

### 1. **Understanding the Gauss-Legendre Quadrature:**
The two-point Gauss-Legendre quadrature formula for approximating an integral is given by:

\[
\int_{a}^{b} f(x) \, dx \approx \frac{b-a}{2} \left[ w_1 f\left(x_1\right) + w_2 f\left(x_2\right) \right]
\]

where:
- \( w_1 = w_2 = 1 \) are the weights for the two-point rule,
- \( x_1 = \frac{a+b}{2} - \frac{b-a}{2\sqrt{3}} \),
- \( x_2 = \frac{a+b}{2} + \frac{b-a}{2\sqrt{3}} \).

### 2. **Applying the Gauss-Legendre Rule:**
Let’s consider the integral over the interval \([a, b]\). Since the interval isn’t provided, let's assume the interval is \([a, b]\) and we proceed with the general case.

1. Compute the points:
   \[
   x_1 = \frac{a+b}{2} - \frac{b-a}{2\sqrt{3}}, \quad x_2 = \frac{a+b}{2} + \frac{b-a}{2\sqrt{3}}
   \]

2. Evaluate the function at these points:
   \[
   f(x_1) = e^{-\cos(x_1)}, \quad f(x_2) = e^{-\cos(x_2)}
   \]

3. Plug the values into the Gauss-Legendre formula:
   \[
   I \approx \frac{b-a}{2} \left[ f(x_1) + f(x_2) \right]
   \]

### 3. **Example Calculation:**
Let's compute a specific example for \( \int_0^\pi e^{-\cos x} \, dx \).

1. Set \( a = 0 \) and \( b = \pi \).
2. Compute the points:
   \[
   x_1 = \frac{\pi}{2} - \frac{\pi}{2\sqrt{3}}, \quad x_2 = \frac{\pi}{2} + \frac{\pi}{2\sqrt{3}}
   \]

3. Evaluate the function:
   \[
   f(x_1) = e^{-\cos\left(\frac{\pi}{2} - \frac{\pi}{2\sqrt{3}}\right)}, \quad f(x_2) = e^{-\cos\left(\frac{\pi}{2} + \frac{\pi}{2\sqrt{3}}\right)}
   \]

4. Plug into the formula:
   \[
   I \approx \frac{\pi - 0}{2} \left[ e^{-\cos\left(\frac{\pi}{2} - \frac{\pi}{2\sqrt{3}}\right)} + e^{-\cos\left(\frac{\pi}{2} + \frac{\pi}{2\sqrt{3}}\right)} \right]
   \]

This would give the approximate value of the integral.

### 4. **General Approach:**
If you’re solving for a general interval \([a, b]\), you can use the points \(x_1\) and \(x_2\) as given and follow the above steps to compute the integral approximation.

### Do you have any questions or would you like further details?

**Related Questions:**

1. How do you perform Gauss-Legendre quadrature with more points?
2. What are the properties of the Legendre polynomials used in Gauss quadrature?
3. How does the Gauss-Legendre rule compare with Simpson's rule in terms of accuracy?
4. What happens if we apply the Gauss-Legendre rule to a non-polynomial function?
5. How can the Gauss-Legendre quadrature be generalized to higher dimensions?

**Tip:** The Gauss-Legendre quadrature is highly accurate for polynomials of degree up to \(2n-1\), where \(n\) is the number of points used.

Learn how to compute the integral of e^-cos(x) using the two-point Gauss-Legendre quadrature rule. This problem involves numerical integration techniques and the application of specific mathematical formulas for accurate approximation. Suitable for graduate-level students and researchers.

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