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.