ImproperIntegralExamples 2009-04-18 08:24:10 2009-04-18 08:24:10 Topic synon. % this is the default PlanetPhysics preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \PMlinkid{1.}{7065} \; $\displaystyle\int_0^\infty e^{-x^2}\,dx \;=\; \frac{\sqrt{\pi}}{2}$\\ \PMlinkid{2.}{11495} \; $\displaystyle\int_0^\infty e^{-x^2}\cos{kx}\,dx\;=\;\frac{\sqrt{\pi}}{2}e^{-\frac{1}{4}k^2}$\\ \PMlinkid{3.}{11504} \; $\displaystyle\int_0^\infty \frac{e^{-x^2}}{a^2\!+\!x^2}\,dx \;=\;\frac{\pi}{2a}e^{a^2}\,{\rm erfc}\,a$\\ \PMlinkid{4.}{10980} \; $\displaystyle\int_0^\infty\sin{x^2}\,dx \;=\; \int_0^\infty\cos{x^2}\,dx \;=\; \frac{\sqrt{2\pi}}{4}$\\ \PMlinkid{5.}{7082} \; $\displaystyle\int_0^\infty\frac{\sin{ax}}{x}\,dx \;=\; (\mbox{sgn}\,a)\frac{\pi}{2} \qquad (a \in \mathbb{R})$\\ \PMlinkid{6.}{11487} \; $\displaystyle\int_0^\infty\left(\frac{\sin{x}}{x}\right)^2 dx \;=\; \frac{\pi}{2}$\\ \PMlinkid{7.}{11487} \; $\displaystyle\int_0^\infty\frac{1-\cos{kx}}{x^2}\,dx \;=\; \frac{\pi k}{2}$\\ \PMlinkid{8.}{11480} \; $\displaystyle\int_0^\infty\frac{x^{-k}}{x\!+\!1}\,dx \;=\; \frac{\pi}{\sin{\pi k}} \quad (0 < k < 1)$\\ \PMlinkid{9.}{11544} \; $\displaystyle\int_{-\infty}^\infty\frac{e^{kx}}{1\!+\!e^x}\,dx \;=\; \frac{\pi}{\sin{\pi k}} \quad (0 < k < 1)$\\ \PMlinkid{10.}{7136} \; $\displaystyle\int_0^\infty\frac{\cos{kx}}{x^2\!+\!1}\,dx \;=\; \frac{\pi}{2e^k}$\\ \PMlinkid{11.}{11489} \; $\displaystyle\int_0^\infty\frac{a\cos{x}}{x^2\!+\!a^2}\,dx \;=\; \int_0^\infty\frac{x\sin{x}}{x^2\!+\!a^2}\,dx \;=\; \frac{\pi}{2e^a} \quad\; (a > 0)$\\ \PMlinkid{12.}{11547} \; $\displaystyle\int_0^\infty\frac{\sin{ax}}{x(x^2\!+\!1)}\,dx \;=\; \frac{\pi}{2}(1-e^{-a}) \quad\; (a > 0)$\\ \PMlinkid{13.}{9223} \; $\displaystyle\int_0^\infty e^{-x}x^{-\frac{3}{2}}\,dx \;=\; \sqrt{\pi}$\\ \PMlinkid{14.}{10637} \; $\displaystyle\int_0^\infty e^{-x}x^3\sin{x}\,dx \;=\; 0$\\ \PMlinkid{15.}{7891} \; $\displaystyle\int_0^\infty\!\left(\frac{1}{e^x\!-\!1}-\frac{1}{xe^x}\right) dx \;=\; \gamma$\\ \PMlinkid{16.}{11516} \; $\displaystyle\int_0^\infty\!\frac{\cos{ax^2}-\cos{ax}}{x} dx \;=\; \frac{\gamma+\ln{a}}{2} \quad (a > 0)$\\ \PMlinkid{17.}{11511} \; $\displaystyle\int_0^\infty\frac{e^{-ax}\!-\!e^{-bx}}{x}\,dx \;=\; \ln\frac{b}{a} \quad (a > 0,\;\, b > 0)$\\ \PMlinkid{18.}{11526} \; $\displaystyle\int_1^\infty\left(\arcsin\frac{1}{x}-\frac{1}{x}\right)\,dx \;=\; 1+\ln{2}-\frac{\pi}{2}$\\ \PMlinkid{19.}{11617} \; $\displaystyle\int_0^1\frac{\arctan{x}}{x\sqrt{1\!-\!x^2}}\,dx \;=\; \frac{\pi}{2}\ln(1\!+\!\sqrt{2})$\\ Link to the \PMlinkexternal{original entry}{http://planetmath.org/encyclopedia/ListOfImproperIntegrals.html} from which one can find the derivations of the given values.