How to apply Gaussian quadrature formula

Question

Evaluate $$ \int_0^{\infty} e^{-x}\ dx $$ employing three points Gaussian quadrature formula, finding the required weights and residues. Use five decimal places for computation.

How to change integral limits to -1 to 1?

Answer 1

$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,} \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack} \newcommand{\dd}{\mathrm{d}} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,} \newcommand{\ic}{\mathrm{i}} \newcommand{\mc}[1]{\mathcal{#1}} \newcommand{\mrm}[1]{\mathrm{#1}} \newcommand{\pars}[1]{\left(\,{#1}\,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,} \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}} \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$

With A & S Table $\ds{\mathbf{25.4.45}}$ Formula which is Gauss-Laguerre Quadrature Formula:

\begin{align} \int_{0}^{\infty}\expo{-x}\dd x & \approx w_{1} + w_{2} + w_{3} = 1.0000003\,,\qquad\qquad\qquad \left\{\begin{array}{rcl} \ds{w_{1}} & \ds{=} & \ds{0.0103893} \\ \ds{w_{2}} & \ds{=} & \ds{0.711093} \\ \ds{w_{3}} & \ds{=} & \ds{0.278518} \end{array}\right. \end{align}

The Relative Error becomes $\ds{\verts{1.0000003 - 1 \over 1}\ 100\ \% = \bbx{3 \times 10^{-5}\ \%}}$ !!!.