Gaussian Quadrature Calculator

Numerically evaluate definite integrals with Gauss quadrature: Legendre, Chebyshev, Laguerre, or Hermite. Enter f(x) and the order to get the integral plus nodes and weights.

Input

Enter the integrand f(x), a quadrature rule, and the order to numerically evaluate the integral by Gaussian quadrature.

Integrand f(x)

e.g. exp(-x^2), sin(x)/x, 2x+1. Supports functions like sin, cos, tan, exp, log, ln, sqrt, abs, the constants pi and e, and implicit multiplication.

Quadrature rule

Computes ∫ₐᵇ f(x) dx (weight function w(x)=1).

Lower limit a

Upper limit b

Order n (number of nodes)

An integer from 1 to 64. Larger values give higher accuracy.

Result

Integral value (approx.)

0.7468241268

Rule

Gauss-Legendre (interval [a, b])

Order n

Nodes and weights

i	Node xᵢ	Weight wᵢ
1	0.046910077	0.2369268851
2	0.2307653449	0.4786286705
3	0.5	0.5688888889
4	0.7692346551	0.4786286705
5	0.953089923	0.2369268851

How it works

Gaussian quadrature approximates an integral by evaluating the integrand at the roots (nodes) of an order-n orthogonal polynomial and summing the values weighted by the corresponding weights. An n-point rule integrates polynomials up to degree 2n-1 exactly.
The Legendre rule uses weight function w(x)=1 and maps any interval [a,b] onto [-1,1] to compute ∫ f(x) dx.
The Chebyshev rule uses weight function w(x)=1/√(1-x²) on [-1,1] and computes ∫ f(x)/√(1-x²) dx; its nodes and weights have closed-form expressions.
The Laguerre rule uses weight w(x)=e^(-x) on [0,∞) to compute ∫ e^(-x) f(x) dx, and the Hermite rule uses weight w(x)=e^(-x²) on (-∞,∞) to compute ∫ e^(-x²) f(x) dx.
Nodes and weights are obtained as the eigenvalues and eigenvectors of the tridiagonal matrix built from the three-term recurrence coefficients (independent implementation). f(x) is parsed by a built-in expression parser supporting functions such as sin, cos, exp, log, sqrt, the constants π and e, and implicit multiplication (e.g. 2x).
Results are numerical approximations. Errors may appear depending on the integrand and the chosen order.

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