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单词 TwoImproperIntegrals
释义

two improper integrals


Let us consider first the improper integral

I(k):=01-coskxx2𝑑x.

The derivativeMathworldPlanetmathPlanetmath I(k) may be formed by differentiating under the integral sign (http://planetmath.org/DifferentiationUnderIntegralSign):

I(k)=0(k1-coskxx2)𝑑x=0sinkxx𝑑x=0sintt𝑑t

Here, the last form has been gotten by the substitution (http://planetmath.org/ChangeOfVariableInDefiniteIntegral)  kx=t. But since by the parent entry (http://planetmath.org/SineIntegralInInfinity) we have

0sintt𝑑t=π2

and since  I(0)=0, we can write

I(k)=0kπ2𝑑k=πk2.

Thus we have evaluated the integral I(k):

01-coskxx2𝑑x=πk2.(1)

The formulaMathworldPlanetmathPlanetmath (1) gives

I(1)=01-cosxx2𝑑x=π2.

We use here the consequence formula

1-cosx= 2sin2x2

of the double angle formula  cos2α=1-2sin2α,  obtaining

π2= 20sin2x2x2𝑑x=0sin2uu2𝑑u,

where the substitution  x2=u  has produced the last form.  Accordingly, we can write as result the formula

0(sinxx)2𝑑x=π2.(2)
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更新时间:2025/5/5 2:53:31