Gauss-Chebyshev Quadrature (First Kind) Calculator

Enter an integrand f(x) and order n to compute the weighted integral ∫[-1,1] f(x)/√(1−x²) dx by Gauss-Chebyshev quadrature of the first kind, with the nodes, weights, and function values listed.

Input

Enter the integrand f(x) and the order n to compute the weighted integral ∫[-1,1] f(x)/√(1−x²) dx by first-kind Gauss-Chebyshev quadrature. The weight 1/√(1−x²) is handled separately, so do not include it in f(x).

Integrand f(x)

e.g. exp(x), cos(x), x^2 + 1. Do not include the weight 1/√(1−x²). See the notes for available functions, constants, and operators.

Order n

Number of nodes (1–256)

Result

Integral ∫ f(x)/√(1−x²) dx

3.9774632605

Interval [-1, 1] with weight 1/√(1−x²)

Order n

Nodes

Weight w_i = π/n

0.39269908

Nodes, weights, and function values

Shows the nodes x_i = cos((2i−1)π/2n), weights w_i = π/n, and function values f(x_i).

#	Node x_i	Weight w_i	f(x_i)
1	-0.98078528	0.39269908	0.37501649
2	-0.83146961	0.39269908	0.43540893
3	-0.55557023	0.39269908	0.573745
4	-0.19509032	0.39269908	0.82276034
5	0.19509032	0.39269908	1.21542076
6	0.55557023	0.39269908	1.74293458
7	0.83146961	0.39269908	2.29669151
8	0.98078528	0.39269908	2.66654941

How it works

First-kind Gauss-Chebyshev quadrature approximates the weighted integral ∫[-1,1] f(x)/√(1−x²) dx ≈ Σ w_i f(x_i) over [-1, 1] with weight function 1/√(1−x²). The endpoint singularities (x=±1) are absorbed by the quadrature rule.
The nodes (zeros of the first-kind Chebyshev polynomial T_n) are given in closed form by x_i = cos((2i−1)π/(2n)) for i=1,…,n, so no iterative root finding is required.
All weights are equal: w_i = π/n. Hence the approximation is simply (π/n)·Σ f(x_i).
The rule is exact for polynomials f(x) up to degree 2n−1.
Enter f(x) as the integrand excluding the weight function 1/√(1−x²) — do not include 1/√(1−x²) in f(x) yourself.
f(x) supports x with +, −, *, /, ^, functions such as sin, cos, tan, exp, log, sqrt, abs, and the constants pi and e.

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