list of improper integrals
Below, we list some
convergent (http://planetmath.org/ConvergenceOfIntegrals) improper
integrals.
1. (http://planetmath.org/areaundergaussiancurve) ∫∞0e-x2𝑑x=√π2
2. (http://planetmath.org/generalisationofgaussianintegral) ∫∞0e-x2coskxdx=√π2e-14k2
3. (http://planetmath.org/usingconvolutiontofindlaplacetransform) ∫∞0e-x2a2+x2𝑑x=π2aea2erfca
4. (http://planetmath.org/fresnelformulas) ∫∞0sinx2dx=∫∞0cosx2dx=√2π4
5. (http://planetmath.org/sineintegralatinfinity) ∫∞0sinaxxdx=(sgna)π2 (a∈ℝ)
6. (http://planetmath.org/twoimproperintegrals) ∫∞0(sinxx)2𝑑x=π2
7. (http://planetmath.org/twoimproperintegrals) ∫∞01-coskxx2𝑑x=πk2
8. (http://planetmath.org/usingresiduetheoremnearbranchpoint) ∫∞0x-kx+1dx=πsinπk (0<k<1)
9. (http://planetmath.org/exampleofchangingvariable) ∫∞-∞ekx1+exdx=πsinπk (0<k<1)
10. (http://planetmath.org/exampleofusingresiduetheorem) ∫∞0coskxx2+1𝑑x=π2ek
11. (http://planetmath.org/laplaceintegrals) ∫∞0acosxx2+a2dx=∫∞0xsinxx2+a2dx=π2ea (a>0)
12. (http://planetmath.org/applicationofsineintegralatinfinity) ∫∞0sinaxx(x2+1)dx=π2(1-e-a) (a>0)
http://planetmath.org/node/922313. ∫∞0e-xx-32𝑑x=√π
14. (http://planetmath.org/laplacetransformoftnft) ∫∞0e-xx3sinxdx= 0
http://planetmath.org/node/789115. ∫∞0(1ex-1-1xex)𝑑x=γ
16. (http://planetmath.org/relativeofcosineintegral) ∫∞0cosax2-cosaxxdx=γ+lna2 (a>0)
17. (http://planetmath.org/relativeofexponentialintegral) ∫∞0e-ax-e-bxxdx=lnba (a>0,b>0)
18. (http://planetmath.org/integralrelatedtoarcsine) ∫∞1(arcsin1x-1x)𝑑x= 1+ln2-π2
19. (http://planetmath.org/exampleofimproperintegral) ∫10arctanxx√1-x2𝑑x=π2ln(1+√2)=π2arsinh 1
20. (http://planetmath.org/applicationoflogarithmseries)
∫10ln(1+x)x𝑑x=π212
21. ∫112ln(1-x)x2𝑑x=-2ln2
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 |