Chapter 5, Problem 5PS

Approximation The Two-Point Gaussian Quadrature Approximation for f is ${\displaystyle {\int }_{-1}^{1}f(x)dx\approx f\left(-\frac{1}{\sqrt{3}}\right)+f\left(\frac{1}{\sqrt{3}}\right)}$

(a) Use this formula to approximate ${\displaystyle {\int }_{-1}^1\cos xdx}$ Find the error of the approximation.

(b) Use this formula to approximate ${\displaystyle {\int }_{-1}^1\frac{1}{1+x^2}dx}$

(c) Prove that the Two-Point Gaussian Quadrature Approximation is exact for all polynomials of degree 3 or less.