In science there are several important orthogonal polynomials which arise from the solution of differential equations. Well-known examples are the Legendre, Hermite, Laguerre and Chebyshev polynomials. They have the following weight functions
| Weight function | Interval | Polynomial |
| W(x)=1 | x\in [-1,1] | Legendre |
| W(x)=e^{-x^2} | -\infty \le x \le \infty | Hermite |
| W(x)=x^{\alpha}e^{-x} | 0 \le x \le \infty | Laguerre |
| W(x)=1/(\sqrt{1-x^2}) | -1 \le x \le 1 | Chebyshev |
The importance of the use of orthogonal polynomials in the evaluation of integrals can be summarized as follows.