Question

(i) The Legendre polynomial of degree four is P ⁴ (x)=x ⁴−6/7x²+3/35. The roots xᵢ of P₄ (x) and the constants cᵢ =

∫ ¹₋₁ ∏ j=in x-xⱼ/xᵢ-xⱼdx are given in the following table: Apply Gaussian quadrature with n=4 to approximate ∫
∫₅¹sin(x)/x dx
(ii) What is the highest degree polynomial for which Gaussian quadrature with n=4 gives the exact value?

Answer 1

To approximate the integral ∫ ∫₅¹sin(x)/x dx using Gaussian quadrature with n=4, we can use the roots and constants given in the Legendre polynomial table. The highest degree polynomial for which Gaussian quadrature with n=4 gives the exact value is 2.

Step-by-step explanation:

To approximate the integral ∫ ∫₅¹sin(x)/x dx using Gaussian quadrature with n=4, we can use the roots and constants given in the Legendre polynomial table. The roots are x₁ = -0.90618, x₂ = -0.53847, x₃ = 0, and x₄ = 0.53847. The constants are c₁ = 0.23693, c₂ = 0.47863, c₃ = 0.56889, and c₄ = 0.23693.

Using these values, we can calculate the approximation for the integral by summing up the product of the function value at each root and its corresponding constant.

The highest degree polynomial for which Gaussian quadrature with n=4 gives the exact value is 2.