Gaussian integration

The integral over a finite interval of a function $w(t)f(t)$ for an arbitrary function $f$ with a non-negative weight function $w(t)$ can be approximated by integrating its approximation by Lagrange polynomials:

$$\int\limits _{a}^{b}w(t)f(t)dt \approx \int\limits _{a}^{b}w(t)\sum ^{n}_{j=1}\prod ^{n}_{\begin{array}{... ..._{i}}f(t_{j})\mathrm{d}t+\int\limits _{a}^{b}w(t)v(t)f[t_{1},\ldots ,t_{n},t]dt$$

$$= \sum ^{n}_{j=1}\int\limits _{a}^{b}w(t)\prod ^{n}_{\begin{array}{... ...1\\ i\neq j \end{array}}\frac{t-t_{i}}{t_{j}-t_{i}}\mathrm{d}t\, f(t_{j})+R(f)$$

$$= \sum _{j=1}^{n}A_{j}f(t_{j})+R(f)$$ (12)

where $v(t)=\prod\limits _{i=1}^{n}(t-t_{i})$. Thus the constant coefficients

$$A_{j}=\int\limits _{a}^{b}w(t)\prod ^{n}_{\begin{array}{c} i=1\\ i\neq j \end{array}}\frac{t-t_{i}}{t_{j}-t_{i}}\mathrm{d}t$$

are found which are used for the approximation of the integral.