Find the integral I = integral ofe^-x^2 cos xdx using two-point Gauss Legendre rule. Marks

To evaluate the integral \( I = \int e^{-x^2} \cos(x) \, dx \) using the two-point Gauss-Legendre quadrature rule, we need to follow these steps:

### 1. **Gauss-Legendre Quadrature Rule:**
The Gauss-Legendre quadrature rule approximates an integral of the form:

\[
\int_{-1}^{1} f(x) \, dx \approx \sum_{i=1}^{n} w_i f(x_i)
\]

where \( x_i \) are the roots of the Legendre polynomial \( P_n(x) \), and \( w_i \) are the corresponding weights.

For the two-point rule (\( n = 2 \)):

- The roots of the Legendre polynomial \( P_2(x) \) are \( x_1 = -\frac{1}{\sqrt{3}} \) and \( x_2 = \frac{1}{\sqrt{3}} \).
- The weights are \( w_1 = w_2 = 1 \).

### 2. **Change of Interval:**
If the integral is over a different interval \([a, b]\), we first need to map it to \([-1, 1]\) using the substitution:

\[
x = \frac{b-a}{2} t + \frac{b+a}{2}
\]

where \( t \) ranges over \([-1, 1]\).

However, since the integral in this problem is indefinite (no limits are given), you may need to consider limits or evaluate the integral over a specific interval.

### 3. **Applying the Gauss-Legendre Rule:**
Let's assume the integral is over a symmetric interval, say \([-1, 1]\), for simplicity.

So we approximate:

\[
I \approx \frac{b-a}{2} \sum_{i=1}^{2} w_i \cdot f\left( \frac{b-a}{2} x_i + \frac{b+a}{2} \right)
\]

Let's assume \( a = -1 \) and \( b = 1 \).

For the function \( f(x) = e^{-x^2} \cos(x) \), we have:

\[
I \approx \sum_{i=1}^{2} w_i \cdot e^{-\left( \frac{1}{\sqrt{3}} \right)^2} \cos\left( \frac{1}{\sqrt{3}} \right)
\]

### 4. **Evaluation:**
Substituting the values \( x_1 = -\frac{1}{\sqrt{3}} \) and \( x_2 = \frac{1}{\sqrt{3}} \):

\[
I \approx 1 \cdot e^{-\left( \frac{1}{\sqrt{3}} \right)^2} \cos\left( -\frac{1}{\sqrt{3}} \right) + 1 \cdot e^{-\left( \frac{1}{\sqrt{3}} \right)^2} \cos\left( \frac{1}{\sqrt{3}} \right)
\]

Given that \( \cos(x) \) is an even function:

\[
I \approx 2 \cdot e^{-\frac{1}{3}} \cos\left( \frac{1}{\sqrt{3}} \right)
\]

This expression provides an approximation to the integral \( I \).

### 5. **Conclusion:**
The integral \( I \) can be approximately evaluated using the two-point Gauss-Legendre quadrature as:

\[
I \approx 2 \cdot e^{-\frac{1}{3}} \cos\left( \frac{1}{\sqrt{3}} \right)
\]

Would you like a further breakdown of these steps or more details? 

Here are some related questions for further exploration:

1. How would the result change if we used a higher-order Gauss-Legendre quadrature rule?
2. How do we handle integrals with infinite limits using Gauss-Legendre quadrature?
3. What are the roots and weights for the three-point Gauss-Legendre quadrature rule?
4. How does the accuracy of the Gauss-Legendre method compare to other numerical integration techniques?
5. What is the significance of the Legendre polynomials in numerical integration?

**Tip:** When applying numerical integration methods like Gauss-Legendre quadrature, ensure that the integral's limits are properly transformed to match the method's interval, usually \([-1, 1]\).

Learn how to evaluate the integral of e^-x^2 cos xdx using the two-point Gauss-Legendre quadrature rule with a detailed step-by-step solution. This problem involves numerical integration methods and the application of Gauss-Legendre quadrature, suitable for advanced undergraduate students.

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