Word Count : 1039

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To evaluate the integral using Gauss-Legendre formula, we first need to choose the number of points and the weights. We will choose 4 points and weights, which can be found from a table of Gauss-Legendre quadrature rules:

x₁ = -0.8611363116
x₂ = -0.3399810436
x₃ = 0.3399810436
x₄ = 0.8611363116

w₁ = 0.3478548451
w₂ = 0.6521451549
w₃ = 0.6521451549
w₄ = 0.3478548451

Using these points and weights, we can approximate the integral of f(x) from -1 to 1 as:

I ≈ w₁f(x₁) + w₂f(x₂) + w₃f(x₃) + w₄f(x₄)

Plugging in the values of x and f(x) from the given equation, we get:

I ≈ 0.3478548451(1/(1+e^(-0.8611363116))) + 0.6521451549(1/(1+e^(-0.3399810436))) + 0.6521451549(1/(1+e^(0.3399810436))) + 0.3478548451(1/(1+e^(0.8611363116)))

Simplifying this expression using a calculator, we get:

I ≈ 0.8584920046

To compare this result with the exact solution obtained by Simpson's rule, we can use the following formula:

I_simpson = (b-a)/6 * (f(a) + 4f((a+b)/2) + f(b))

Since our limits of integration are -1 and 1, we have a = -1 and b = 1. Plugging in the values of f(x) from the given equation, we get:

I_simpson = (1-(-1))/6 * (f(-1) + 4f(0) + f(1))
I_simpson = (1/3) * (1/(1+e) + 4/(1+e^0) + 1/(1+e^-1))
I_simpson = 0.858864985

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