# methods of evaluating improper integrals

There are some general methods of evaluating improper integrals in such cases when one cannot directly use the antiderivative of the integrand.  Which method is suitable in a certain instance, is dependent on the kind of the integral (http://planetmath.org/DefiniteIntegral).

• Differentiation under the integral sign with respect to a parametre in the integrand; one can add a new parametre to a suitable place.  The differentiated form may then be integrated directly or from a differential equation.  Examples: a (http://planetmath.org/twoimproperintegrals), b (http://planetmath.org/generalisationofgaussianintegral), c (http://planetmath.org/relativeofexponentialintegral), d (http://planetmath.org/integralrelatedtoarcsine), http://planetmath.org/node/11617e.

• Laplace transform (http://planetmath.org/laplacetransform).  If the integrand has, as above, a parametre in a suitable place, the Laplace transform of the integrand with respect to this parametre is often simpler to integrate and the new improper integral to evaluate; thereafter one simply inversely.  Examples: f (http://planetmath.org/sineintegralatinfinity), g (http://planetmath.org/usingconvolutiontofindlaplacetransform), h (http://planetmath.org/relativeofcosineintegral), i (http://planetmath.org/laplacetransformoftnft), http://planetmath.org/node/10637j.

• Cauchy residue theorem.  The integral may be obtained as limit of a contour integral in the complex plane.  Examples: k (http://planetmath.org/usingresiduetheoremnearbranchpoint), l (http://planetmath.org/fresnelformulas), m (http://planetmath.org/laplaceintegrals), http://planetmath.org/node/11489n.

• Expanding the integrand to series.  Example: o (http://planetmath.org/applicationoflogarithmseries).

• Changing variable (http://planetmath.org/ChangeOfVariableInDefiniteIntegral) in an improper integral sometimes may recur it to a known improper integral.  Examples: p (http://planetmath.org/areaundergaussiancurve), q (http://planetmath.org/exampleofimproperintegral), r (http://planetmath.org/exampleofchangingvariable).

Title methods of evaluating improper integrals MethodsOfEvaluatingImproperIntegrals 2014-11-07 19:12:16 2014-11-07 19:12:16 pahio (2872) pahio (2872) 34 pahio (2872) Application msc 40A10 IntegrationOfLaplaceTransformWithRespectToParameter ListOfImproperIntegrals IntegralRelatedToArcSine ExampleOfChangingVariable