list of improper integrals

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

1. (http://planetmath.org/areaundergaussiancurve)   ${\displaystyle {\int }_0^{\mathrm{\infty }}}{e}^{-x^2}𝑑x={\displaystyle \frac{\sqrt{\pi }}{2}}$

2. (http://planetmath.org/generalisationofgaussianintegral)   ${\displaystyle {\int }_0^{\mathrm{\infty }}}{e}^{-x^2}\cos kxdx={\displaystyle \frac{\sqrt{\pi }}{2}}{e}^{-\frac{1}{4}{k}^2}$

3. (http://planetmath.org/usingconvolutiontofindlaplacetransform)   ${\displaystyle {\int }_0^{\mathrm{\infty }}}{\displaystyle \frac{e^{-x^2}}{a^2+x^2}}𝑑x={\displaystyle \frac{\pi }{2a}}{e}^{a^2}\mathrm{erfc}a$

4. (http://planetmath.org/fresnelformulas)   ${\displaystyle {\int }_0^{\mathrm{\infty }}}\sin x^{2}dx={\displaystyle {\int }_0^{\mathrm{\infty }}}\cos x^{2}dx={\displaystyle \frac{\sqrt{2\pi }}{4}}$

5. (http://planetmath.org/sineintegralatinfinity)   ${\displaystyle {\int }_0^{\mathrm{\infty }}}{\displaystyle \frac{\sin ax}{x}}dx=(\text{sgn}a){\displaystyle \frac{\pi }{2}}\mathit{\hspace{1em}\hspace{1em}}(a\in ℝ)$

6. (http://planetmath.org/twoimproperintegrals)   ${\displaystyle {\int }_0^{\mathrm{\infty }}}{\left({\displaystyle \frac{\sin x}{x}}\right)}^2𝑑x={\displaystyle \frac{\pi }{2}}$

7. (http://planetmath.org/twoimproperintegrals)   ${\displaystyle {\int }_0^{\mathrm{\infty }}}{\displaystyle \frac{1-\cos kx}{x^2}}𝑑x={\displaystyle \frac{\pi k}{2}}$

8. (http://planetmath.org/usingresiduetheoremnearbranchpoint)  

9. (http://planetmath.org/exampleofchangingvariable)  

10. (http://planetmath.org/exampleofusingresiduetheorem)   ${\displaystyle {\int }_0^{\mathrm{\infty }}}{\displaystyle \frac{\cos kx}{x^2+1}}𝑑x={\displaystyle \frac{\pi }{2e^k}}$

11. (http://planetmath.org/laplaceintegrals)   ${\displaystyle {\int }_0^{\mathrm{\infty }}}{\displaystyle \frac{a\cos x}{x^2+a^2}}dx={\displaystyle {\int }_0^{\mathrm{\infty }}}{\displaystyle \frac{x\sin x}{x^2+a^2}}dx={\displaystyle \frac{\pi }{2e^a}}\mathit{\hspace{1em}\hspace{0.25em}}(a>0)$

12. (http://planetmath.org/applicationofsineintegralatinfinity)   ${\displaystyle {\int }_0^{\mathrm{\infty }}}{\displaystyle \frac{\sin ax}{x(x^2+1)}}dx={\displaystyle \frac{\pi }{2}}(1-e^{-a})\mathit{\hspace{1em}\hspace{0.25em}}(a>0)$

http://planetmath.org/node/922313.   ${\displaystyle {\int }_0^{\mathrm{\infty }}}{e}^{-x}{x}^{-\frac{3}{2}}𝑑x=\sqrt{\pi }$

14. (http://planetmath.org/laplacetransformoftnft)   ${\displaystyle {\int }_0^{\mathrm{\infty }}}{e}^{-x}{x}^3\sin xdx=\mathrm{\hspace{0.33em}0}$

http://planetmath.org/node/789115.   ${\displaystyle {\int }_0^{\mathrm{\infty }}}\left({\displaystyle \frac{1}{e^x-1}}-{\displaystyle \frac{1}{x{e}^x}}\right)𝑑x=\gamma$

16. (http://planetmath.org/relativeofcosineintegral)   ${\displaystyle {\int }_0^{\mathrm{\infty }}}{\displaystyle \frac{\cos a{x}^2-\cos ax}{x}}dx={\displaystyle \frac{\gamma +\ln a}{2}}\mathit{\hspace{1em}}(a>0)$

17. (http://planetmath.org/relativeofexponentialintegral)   ${\displaystyle {\int }_0^{\mathrm{\infty }}}{\displaystyle \frac{e^{-ax}-e^{-bx}}{x}}dx=\ln {\displaystyle \frac{b}{a}}\mathit{\hspace{1em}}(a>0,b>0)$

18. (http://planetmath.org/integralrelatedtoarcsine)   ${\displaystyle {\int }_1^{\mathrm{\infty }}}\left(\arcsin {\displaystyle \frac{1}{x}}-{\displaystyle \frac{1}{x}}\right)𝑑x=\mathrm{\hspace{0.33em}1}+\ln 2-{\displaystyle \frac{\pi }{2}}$

19. (http://planetmath.org/exampleofimproperintegral)   ${\displaystyle {\int }_0^1}{\displaystyle \frac{\arctan x}{x\sqrt{1-x^2}}}𝑑x={\displaystyle \frac{\pi }{2}}\ln (1+\sqrt{2})={\displaystyle \frac{\pi }{2}}\text{arsinh}\mathrm{\hspace{0.17em}1}$

20. (http://planetmath.org/applicationoflogarithmseries)  ${\displaystyle {\int }_0^1}{\displaystyle \frac{\ln (1+x)}{x}}𝑑x={\displaystyle \frac{{\pi }^2}{12}}$

21.   ${\displaystyle {\int }_{\frac{1}{2}}^1}{\displaystyle \frac{\ln (1-x)}{x^2}}𝑑x=-2\ln 2$

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