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单词 RelativeOfCosineIntegral
释义

relative of cosine integral


For determining of the value of the improper integral

I(a):=0cosax2-cosaxxdx  (a>0),

related to the cosine integralDlmfDlmfDlmfMathworldPlanetmath, we think it as a functionMathworldPlanetmath of the parametre a which we denote by t.  Then we can take the Laplace transformDlmfMathworldPlanetmath (see the integration with respect to a parametre in the table of Laplace transforms):

{I(t)}={0(costx2-costx)dxx}=0(ss2+x4-ss2+x2)dxx

Splitting the fractional expressions to http://planetmath.org/node/5812partial fractions and integrating give

{I(t)}=1s0(1x-x3s2+x4-1x+xs2+x2)𝑑x
=1s/x=0[12ln(s2+x2)-14ln(s2+x4)]
=14/x=0ln(s2+x2)2s2+x4=-lns2s.

As seen in the http://planetmath.org/node/10588table of Laplace transforms, the gotten expression is the Laplace transform of γ+lnt2=I(t)  (N.B.  {1}=1s), and thus we have the result

I(a)=γ+lna2.
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更新时间:2025/5/4 8:48:25