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).
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Differentiation
under the integral sign with respect to aparametre in the integrand; one can add a new parametre to asuitable place. The differentiated form may then beintegrated directly or from a differential equation
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Laplace transform (http://planetmath.org/laplacetransform). If theintegrand has, as above, a parametre in a suitable place, theLaplace transform of the integrand with respect to thisparametre is often simpler to integrate and the new improperintegral 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.
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Cauchy residue theorem. The integral may be obtained aslimit 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.
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Expanding the integrand to series. Example:o (http://planetmath.org/applicationoflogarithmseries).
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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).