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This method is used to calculate a certain integral. It is an approximation of the definite integral of a function, it is usually stated as a weighted sum of function values at specified points in the domain of integration. It is named after Carl Friedrich Gauss which is a quadrature rule developed to yield an exact result of polynomials of degree 2n - 1 or less by choosing a suitable nodes ${\mathrm{x}}_{\mathrm{i}}$ and weights ${\mathrm{w}}_{\mathrm{i}}$ for i = 1,…,n.

DERIVATION

Here W(x) = 1 so the integral to be computed is 

After making the linear transformation

the interval [a, b] transforms to [-1, 1] and setting.

Where

The given integral can be transformed to 

In general, for computing the integral by a quadrature formula with nodes ${\mathrm{t}}_0, {\mathrm{t}}_1, ..., {\mathrm{t}}_{\mathrm{n}}$ having degrees of precision 2n + 1, 

Where ${\mathbf{P}}_{\mathbf{n}}$ is Legendre polynomial of degree n i.e. ${\mathbf{t}}_{\mathbf{0}}\mathbf{,}\mathbf{ }{\mathbf{t}}_{\mathbf{1}}\mathbf{,}\mathbf{ }\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}\mathbf{ }{\mathbf{t}}_{\mathbf{n}}$ are the zero of the polynomial ${\mathbf{P}}_{\mathbf{n}\mathbf{+}\mathbf{1}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}$ and ${\mathbf{c}}_{\mathbf{n}}$ is a constant given by

Hence,

Where

GRAPHICAL REPRESENTATION

INPUT: 

A function for which the integration is to be calculated

OUTPUT: 

The value of integration

PROCESS:

Step 1: [taking the inputs from the user]
	Read n [the number of terms]
	for i = 0 to n - 1 repeat
		Read t[i]
	[End of ‘for’ loop]
	for i = 0 to n - 1 repeat
		Read k[i]
	[End of ‘for’ loop]

Step 2: [Gauss Quadrature]
	Set sum<-0
	for i = 0 to n - 1 repeat
		Set xi ← ((4.5 + 2.5) + (4.5 - 2.5) × t[i])/2
		Set fi ← f(xi)
		Set c ← k[i] × fi
		Set sum ← sum + c
		Print t[i], xi, fi, k[i], c
	[End of ‘for’ loop]
	[‘f(x)’ is a function defined in step 3]
	Set ig ← ((4.5 - 2.5) × sum)/2
	Print ig
[End of Gauss Quadrature]

Step 3: [function f(x)]
	Return exponential(x)/(x^3 - 1)