# list of improper integrals

Below, we list some convergent (http://planetmath.org/ConvergenceOfIntegrals) improper integrals.

1. (http://planetmath.org/areaundergaussiancurve)   $\displaystyle\int_{0}^{\infty}e^{-x^{2}}\,dx\;=\;\frac{\sqrt{\pi}}{2}$

2. (http://planetmath.org/generalisationofgaussianintegral)   $\displaystyle\int_{0}^{\infty}e^{-x^{2}}\cos{kx}\,dx\;=\;\frac{\sqrt{\pi}}{2}e% ^{-\frac{1}{4}k^{2}}$

3. (http://planetmath.org/usingconvolutiontofindlaplacetransform)   $\displaystyle\int_{0}^{\infty}\frac{e^{-x^{2}}}{a^{2}\!+\!x^{2}}\,dx\;=\;\frac% {\pi}{2a}e^{a^{2}}\,{\rm erfc}\,a$

4. (http://planetmath.org/fresnelformulas)   $\displaystyle\int_{0}^{\infty}\sin{x^{2}}\,dx\;=\;\int_{0}^{\infty}\cos{x^{2}}% \,dx\;=\;\frac{\sqrt{2\pi}}{4}$

5. (http://planetmath.org/sineintegralatinfinity)   $\displaystyle\int_{0}^{\infty}\frac{\sin{ax}}{x}\,dx\;=\;(\mbox{sgn}\,a)\frac{% \pi}{2}\qquad(a\in\mathbb{R})$

6. (http://planetmath.org/twoimproperintegrals)   $\displaystyle\int_{0}^{\infty}\left(\frac{\sin{x}}{x}\right)^{2}dx\;=\;\frac{% \pi}{2}$

7. (http://planetmath.org/twoimproperintegrals)   $\displaystyle\int_{0}^{\infty}\frac{1-\cos{kx}}{x^{2}}\,dx\;=\;\frac{\pi k}{2}$

8. (http://planetmath.org/usingresiduetheoremnearbranchpoint)   $\displaystyle\int_{0}^{\infty}\frac{x^{-k}}{x\!+\!1}\,dx\;=\;\frac{\pi}{\sin{% \pi k}}\quad(0

9. (http://planetmath.org/exampleofchangingvariable)   $\displaystyle\int_{-\infty}^{\infty}\frac{e^{kx}}{1\!+\!e^{x}}\,dx\;=\;\frac{% \pi}{\sin{\pi k}}\quad(0

10. (http://planetmath.org/exampleofusingresiduetheorem)   $\displaystyle\int_{0}^{\infty}\frac{\cos{kx}}{x^{2}\!+\!1}\,dx\;=\;\frac{\pi}{% 2e^{k}}$

11. (http://planetmath.org/laplaceintegrals)   $\displaystyle\int_{0}^{\infty}\frac{a\cos{x}}{x^{2}\!+\!a^{2}}\,dx\;=\;\int_{0% }^{\infty}\frac{x\sin{x}}{x^{2}\!+\!a^{2}}\,dx\;=\;\frac{\pi}{2e^{a}}\quad\;(a% >0)$

12. (http://planetmath.org/applicationofsineintegralatinfinity)   $\displaystyle\int_{0}^{\infty}\frac{\sin{ax}}{x(x^{2}\!+\!1)}\,dx\;=\;\frac{% \pi}{2}(1-e^{-a})\quad\;(a>0)$

http://planetmath.org/node/922313.   $\displaystyle\int_{0}^{\infty}e^{-x}x^{-\frac{3}{2}}\,dx\;=\;\sqrt{\pi}$

14. (http://planetmath.org/laplacetransformoftnft)   $\displaystyle\int_{0}^{\infty}e^{-x}x^{3}\sin{x}\,dx\;=\;0$

http://planetmath.org/node/789115.   $\displaystyle\int_{0}^{\infty}\!\left(\frac{1}{e^{x}\!-\!1}-\frac{1}{xe^{x}}% \right)dx\;=\;\gamma$

16. (http://planetmath.org/relativeofcosineintegral)   $\displaystyle\int_{0}^{\infty}\!\frac{\cos{ax^{2}}-\cos{ax}}{x}dx\;=\;\frac{% \gamma+\ln{a}}{2}\quad(a>0)$

17. (http://planetmath.org/relativeofexponentialintegral)   $\displaystyle\int_{0}^{\infty}\frac{e^{-ax}\!-\!e^{-bx}}{x}\,dx\;=\;\ln\frac{b% }{a}\quad(a>0,\;\,b>0)$

18. (http://planetmath.org/integralrelatedtoarcsine)   $\displaystyle\int_{1}^{\infty}\left(\arcsin\frac{1}{x}-\frac{1}{x}\right)\,dx% \;=\;1+\ln{2}-\frac{\pi}{2}$

19. (http://planetmath.org/exampleofimproperintegral)   $\displaystyle\int_{0}^{1}\frac{\arctan{x}}{x\sqrt{1\!-\!x^{2}}}\,dx\;=\;\frac{% \pi}{2}\ln(1\!+\!\sqrt{2})\;=\;\frac{\pi}{2}\,\mbox{arsinh}\,1$

20. (http://planetmath.org/applicationoflogarithmseries)   $\displaystyle\int_{0}^{1}\frac{\ln(1\!+\!x)}{x}\,dx\;=\;\frac{\pi^{2}}{12}$

21.   $\displaystyle\int_{\frac{1}{2}}^{1}\frac{\ln(1\!-\!x)}{x^{2}}\,dx\;=\;-2\ln{2}$

 Title list of improper integrals Canonical name ListOfImproperIntegrals Date of creation 2014-11-07 19:08:22 Last modified on 2014-11-07 19:08:22 Owner pahio (2872) Last modified by pahio (2872) Numerical id 66 Author pahio (2872) Entry type Topic Classification msc 40A10 Related topic ErrorFunction Related topic SignumFunction Related topic EulersConstant Related topic MethodsOfEvaluatingImproperIntegrals Related topic AreaFunctions Related topic ConvergenceOfIntegrals