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单词 IntegrationUnderIntegralSign
释义

integration under integral sign


Let

I(α)=abf(x,α)𝑑x.

where  f(x,α) is continuousMathworldPlanetmath in the rectangle

axb,α1αα2.

Then  αI(α)  is continuous and hence integrable (http://planetmath.org/RiemannIntegrable) on the interval α1αα2;  we have

α1α2I(α)𝑑α=α1α2(abf(x,α)𝑑x)𝑑α.

This is a double integral over a in the xα-plane, whence one can change the order of integration (http://planetmath.org/FubinisTheorem) and accordingly write

α1α2(abf(x,α)𝑑x)𝑑α=ab(α1α2f(x,α)𝑑α)𝑑x.

Thus, a definite integral depending on a parametre may be integrated with respect to this parametre by performing the integration under the integral sign.

Example.  For being able to evaluate the improper integral

I=0e-ax-e-bxxdx  (a>0,b>0),

we may interprete the integrand as a definite integral:

e-ax-e-bxx=/α=bae-αxx=abe-αx𝑑α.

Accordingly, we can calculate as follows:

I=0(abe-αx𝑑α)𝑑x
=ab(0e-αx𝑑x)𝑑α
=ab(/x=0-e-αxα)𝑑α
=ab1α𝑑α=/ablnα
=lnba
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更新时间:2025/5/4 21:16:38