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单词 ExampleOfImproperIntegral
释义

example of improper integral


The integrand of

I=01arctanxx1-x2𝑑x(1)

is undefined both at the lower and the upper limitMathworldPlanetmath.  However, the value of the improper integral exists and may be found via the more general integral

I(y)=01arctanxyx1-x2𝑑x.(2)

Denote the integrand of (2) by  f(x,y).  For any fixed real value y,

f(x,y)O(1) as x0,f(x,y)O(11-x2) as x1,

where the Landau big ordo (http://planetmath.org/formaldefinitionoflandaunotation) notation has been used.  Accordingly, the integralDlmfPlanetmath (2) converges for every y.

The inequalityMathworldPlanetmath

|f(x,y)y|=1(1+x2y2)1-x211-x2

and the convergence of the integral

01dx1-x2=π2

imply that the integral

01f(x,y)y𝑑x(3)

http://planetmath.org/node/6277converges uniformly on the whole y-axis and equals I(y).  For expressing this derivative in a closed formMathworldPlanetmath (http://planetmath.org/ExpressibleInClosedForm), one may utilise the changes of variable (http://planetmath.org/ChangeOfVariableInDefiniteIntegral)

x:=cosφ,tanφ:=t

which yield

I(y)=01dx(1+x2y2)1-x2=0π2dφ1+y2cos2φ
=0dt1+y2+t2=/t=011+y2arctant1+y2
=π21+y2.

Hence,

I(y)=π20ydy1+y2=/0yln(y+1+y2)

and the integral (1) equals  I=I(1)=π2ln(1+2),  i.e.

01arctanxx1-x2𝑑x=π2ln(1+2).(4)
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更新时间:2025/5/4 4:53:01