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单词 IntegralRelatedToArcSine
释义

integral related to arc sine


We want to evaluate the integral

1(arcsin1x-1x)𝑑x.(1)

Therefore we put an extra variable t to the integrand and thus get the function

I(t):=1(arcsintx-tx)𝑑x,

and in to obtain a simpler integral, we differentiate it under the integral sign (http://planetmath.org/DifferentiationUnderIntegralSign), then integrate:

I(t)=1(11-t2x21x-1x)𝑑x
=1(1x2t2-11t-1x)𝑑x
=/x=1[ln(xt+x2t2-1)-lnx]
=/x=1ln1+1-t2x2t
=ln2t-ln1+1-t2t=ln2-ln(1+1-t2)

The gotten expression implies, since  I(0)=1(arcsin0-0)𝑑x=0,  that

I(t)=0t[ln2-ln(1+1-t2)]𝑑t=tln2-0tln(1+1-t2)𝑑t,

and consequently

I(1)=ln2-01ln(1+1-t2)𝑑t=ln2-/01tln(1+1-t2)-01t2dt(1+1-t2)1-t2
=ln2-01t2dt1-t2+1-t2.

Here, the substitution (http://planetmath.org/ChangeOfVariableInDefiniteIntegral)  t=sinu  helps, yielding

I(1)=ln2-0π2(1-cosu)𝑑u=ln2-π2+1.

Accordingly, we have the result

1(arcsin1x-1x)𝑑x= 1+ln2-π2.

For the convergence, see the French version of http://en.wikipedia.org/wiki/Improper_integralthis article.

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更新时间:2025/5/4 16:07:49