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Gaussian Quadrature

​
number of points
2
integrand
sin(x)+3cos(x)
integrand
weights and sampling points
2
∑
i=1
(sin(
x
i
)+3cos(
x
i
))
w
i
= 3.80993
≃
5
2
∫
-2
(sin(x)+3cos(x))x
= 4.9083054917887058962
Gaussian quadrature with
n
points gives exact integral estimates for polynomials of degree
2n-1
or less. For smooth functions, increasing the number of points of the Gaussian rule improves the integral estimate.
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