Chapter 4.3, Problem 70E

To determine

To show: that there exist weights $w_1,w_2,\mathrm{...},w_n$ such that ${\displaystyle \underset{-1}{\overset{1}{\int }}f\left(t\right)dt}={\displaystyle \sum _{i=1}^n{w}_{i}f\left(a_i\right)}$ for all polynomials $f\left(t\right)$ in $P_{n-1}$ .