Chapter 4, Problem 6PS

Approximation TheTwo-Point Gaussian Quadrature Approximation for f is

${\displaystyle {\int }_{-1}^{1}f(x)\approx f\left(-\frac{1}{\sqrt{3}}\right)}+f\left(\frac{1}{\sqrt{3}}\right).$

(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.